The answer, as I see it, is that they didn’t, at least not in Taylor’s ship.  Nor did they return in the ship that we see above.  They couldn't.  Both Taylor's ship and Skipper's ship were total losses, unable to fly ever again.

Jan Rukr's sketch of Brent and / or Skipper's ship with the
fold out landing gear and descent assembly.

Instead, what the the ape-o-nauts returned to Earth's past in was Brent’s ship. The ‘other’ spacecraft that we don’t see in the sequel. So, there are two ANSA spacecraft, following Taylor’s trajectory, and they both encounter the same Hasslein Curve that Taylor's flight ran afoul of.  Right before they make reentry, Brent takes a time reading and gets 3955 AD. The two ANSA spacecraft, this time each is modified to actually make a landing and then a liftoff again, make atmospheric reentry and prepare to land somewhere in the Forbidden Zone because they are tracking an erroneous signal or Landon did somehow get out a signal in the first movie, and these ships picked up that signal after traversing the Hasslein Curve and homed in on it back to this planet.  Maybe Taylor’s spacecraft’s life-raft has drifted away from where they pulled it ashore, and it, like most emergency life rafts today, has a emergency beacon on it so that rescue teams can home in on it. Perhaps the signal source is a poor tribal who has found the deflated raft, and made a spiffy new coat out of it with the beacon still attached and transmitting. This signal is moving across the Forbidden Zone and the rescue party is following what they think is Taylor’s party signal, but is really just a tribal in a bright yellow plastic vest now…

Shane Johnson's conceptual artwork for the design of the ANSA "Tail-Lander" spacecraft.  A much larger image can be found here.

Getting close to the signal, the two ANSA spacecraft, which are 3 seaters, touch down a few hundred or a few thousand feet apart from each other, depending on available terrain (which looks pretty harsh in the picture above), nuclear fusion descent jets firing, landing struts unfolding, only something happens to Skipper’s ship, a major malfunction, and it catches fire or explodes on touch down or shortly thereafter.  Maybe it landed on a rock and bent a landing strut under it (this happened on the moon in real life!), or maybe it landed on ground that was too soft, and as soon as it settled down, one side started leaning over, the strut sinking under the weight of the vehicle, and the whole spacecraft toppled over and exploded...  From the pictures above, it is evident that the ship 'touched down' on its tail (notice the blast pattern from the engine nozzle), caught fire, fell over, and broke apart along major seams / join points.

Here we see Skipper's ship and how it came to rest like it did in the movie.   The ship was making a controlled descent and apparently completed the landing, as evidenced by the scorch pattern beneath the landing gear assembly.  The ship went from position (1) to position (2) in the picture above, settling on its landing gear, scorching the ground under it during landing (or perhaps, if a chemical / nuclear rocket was not used, then the scorch marks would come from whatever explosion occurred in the lower section).  Then we have the malfunction, an explosion or a fire onboard, for whatever reason, and the nearly five story tall ANSA spacecraft collapsed upon itself, moving from an upright position (3) to the horizontal final rest shown in position (4) above.   As it smashed against the embankment, it broke up and 'settled' into its current position.  If the ship had 'slid' into a landing like Alan Virdon's did in the first episode of the TV series (see below), then we would have seen a obvious trench and the nose would have been at the tip of the trench, not rising above the terrain like it is in the picture above.  Why this ship didn't break up upon reentry or hitting the ground would lend credibility to my 'steerable gravity parachute' type of emergency gravity field generator as explained under the MISSION section.

Brent was the first to touch down and is outside of his rescue ship, getting his rescue gear together, trying to get a fix on the moving signal of what they think is Taylor, when he sees Skipper’s spacecraft a good distance away having trouble during its descent.   Unable to help Skipper with his ship by using his own ship, Brent might make contact with Skipper via radio, and when all that can be done has been done, Brent drops the radio, grabs a fire extinguisher, and takes off running for where Skipper's craft is going to land.  Brent manages to run to Skipper’s spacecraft just in time to see the spacecraft superstructure collapse (it was evident that the spacecraft didn’t ‘crash’, it landed (notice the scorch marks from a slow, controlled descent), and witness his commander’s crippled spacecraft topple over, breaking up as it comes crashing down.

It was very evident that something like a fire or an explosion happened, the bottom half of the ANSA spacecraft was obviously weakened or destroyed causing the spacecraft to collapse and topple over. That much is also evident if you look how the vessel is laying and the wreckage is arranged, this spacecraft did not land and plow a furrow like the TV series ANSA spacecraft, this one landed smoothly, exploded, and fell over on its side, crumpling the rest of the superstructure in the process and catching fire.

Brent runs up with a portable fire extinguisher (for all the good that will do) and helps to put out his commander's spacecraft's fire.  Notice the portable fire extinguisher on the outside of the spacecraft, near the wreckage, in the movie?  Why would it be there if the ship had automatic fire extinguishing systems like most military or space vehicles do... ?  I would think that something as arguably complex as these spacecraft are would have something like an onboard fire extinguishing system.  Once the fire is extinguished and the danger is over (and the craft is an obvious loss), Brent pops the locks on the other spacecraft, manually firing the emergency escape hatch, and hauls his mortally wounded commander from the stricken craft.  What we are seeing is not Brent’s fully functional, fully intact, ready to take back off ANSA spacecraft, but Skipper’s doomed and burned one.

The spacecraft fire was very hot, consuming the bottom section before the astronauts could put the blaze out.  Automatic fire control systems were helped with personal fire extinguishers (seen near the wreckage in the movie).  The fire must have had some incredible chemicals or elements involved, such as magnesium, which burned both incredibly hot and incredibly brightly, because it rendered Skipper blind from the glare.  This crew seemed to be more morbid in their realization of their fate as inferred from their dialog after the crash.

... John Brent exits the wreckage of the second ANSA spacecraft.  His wounded commander, "Skipper" tries to rise from where he is laying on a blanket, but Brent forces him gently back down.

Skipper slides into silence and presumably dies shortly thereafter.  Brent then buries Skipper near the wreckage, leaving Brent yet another ANSA astronaut stranded on the Planet of the Apes.  Skipper is the second ANSA astronaut to die during a attempted reentry.  Part of Brent’s desolate outlook on the situation is not only that his commander is gone, but that half of Taylor’s crew just lost their seats for the return flight...  Some rescue mission this is shaping up to be, hmmmm?  Then Brent, alone, knee deep in despair, not used to command but rather used to simply following orders, finds Nova.  Surprised at her appearance, and even more surprised at the dog tags that she has, he gets motivated in a rush of seeing that he might can find Taylor, complete his mission, and also have a commander again.  Taylor will be better at sorting out all of these problems, so Brent hops on the horse with Nova and tells her to take him to Taylor.  Brent, in his haste, leaves his ANSA spacecraft open and unattended.

Brent meets Zira and Cornelius.  The fact that he makes the following statement only seems to prove that he still has a fully functional and operational spacecraft out in the Forbidden Zone.  Brent then says words to the effect of

BRENT:  "I've got to get out of here... I don't know how or what with... but I've got to get back up there."

When Brent says "I've got to get out of here..." he means he has to get out of Ape City and his current situation.  The part where he says "I don't know how or what with..." doesn't mean that he doesn't have a rocket nearby, it just means that he has just been shot, he's in a house with two talking chimpanzees and a cave girl who can't speak, he's surrounded by lots of angry gorillas with guns, and he's running out of ideas.  He got into this situation, but seeing how his way out may be blocked, he's not sure what he's going to use to get himself out of this situation, i.e. knee deep in angry armed gorillas smack dab in the middle of Ape City.

Zack has submitted the following information for us as well:

Oh...I wanted to tell you that I was reading Michael Avallone's novelization of BENEATH and I discovered that Brent had said (when he and Nova were hidden in the brush observing General Ursus' speech in the arena) "I've got to get out," he said, trying to remain levelheaded:

Brent:  "And the only way out--is to take to the sky. I don't know how or what with--all I know is I'm not staying here. If this place has a name, it's the Planet Nightmare..."

This also should prove to doubters that Brent had a ship, too. If Brent and Skipper had arrived in the same ship (which, of course, he couldn't have), he would have had to accept the cold, hard facts that "he was there to stay," like Taylor and his crew did when their ship sank in that lake.

I agree.  Let's look at what Brent says here because it meshes with my own thoughts.  Brent here, in the novel, is saying that the only way to escape is to take to the sky, i.e. use a rocket ship or spacecraft to get away.  The next sentence implies not that he doesn't know how to get back up into the sky, but rather that he feels trapped in Ape City and he doesn't know how or what with, but he has decided not to stay there any longer.  Basically, he's running blind and he's about to seriously start whigging out, if he isn't already doing so.

(And why doesn't he have a gun (Taylor had one!)?  You would assume that if an exploratory mission carried a pistol, that a rescue mission would at least carry one also (probably a good old reliable Colt .45 Government 1911 model) or maybe even a .30 Iver Johnson Carbine or a M16.  You never know what you might find off in space, so the reason why Brent went looking for Taylor unarmed is still a mystery.)

Brent is about to lose it here, folks, he's lost his commander, he's seen stuff that no rational explanation would survive for.  If he can get out of here, out of Ape City, back into the Forbidden Zone and climbing up that ladder to his spacecraft's cockpit, he's almost at the point of saying:

The statement of .".. but I've got to get back up there." clearly shows that there is another spacecraft out there in the Forbidden Zone, a fully functional one ready for lift off.  Who's spacecraft is this?  It isn't Skipper's, we've seen what was left of that one.   This spacecraft has to be Brent's ANSA spacecraft, not the burned and wrecked one of Skipper, and it is ready to go.  All Brent (or three curious monkeys...) has to do is get back to it, throw a few switches, and tell the Planet of the Apes to kiss his lilly white ass as it roars out of the atmosphere, bound for the Hasslein Curve, and on the other side, Disco and the good old US of A.

This statement more than any other shows that Brent has finally, simply wigged out.   He's seen things that his mental psyche just isn't set up to handle.  Time warps, exploding spacecraft, dead commanders, talking apes discussing how to kill humans, and it all rather quickly gets to be too much and Brent cracks a little.  He makes the statement that he doesn't know how he's going to do it, but he's got to get out of Ape City, he's got to get past all of those gorillas with guns and get back to his fully functional spacecraft so he can just get the hell off of the planet and back to the good old U. S. of A. and disco.  Screw Hasslein, screw Taylor, screw these apes, he's gone first chance he gets.  He learns that Landon and Dodge are dead and stuffed in a museum somewhere, and he really wigs out then.  It isn't until he gets information on Taylor that he comes around to his senses again and remembers just why he is here. 

Brent needs a commander, he isn't prepared to make the tough decisions that command requires, he's just not that type of soldier.  When Brent realizes that there still might be a chance to find Taylor (and thus have someone else make all of the big decisions), he resigns himself to be a good soldier and take the mission seriously once again.  He and Nova head off in the other direction into the Forbidden Zone to search for Taylor. Meanwhile, Zira and Cornelius, curious about two talking humans and wondering what brought the humans to their planet, decide to go look in the Forbidden Zone for something that might prove that Taylor and Brent did come from space.  Zira and Cornelius, along with their confidant, Milo, who is more of a bothersome friend than a good friend, and who won't take 'no' for an answer, or maybe is going to tell the gorillas about their little expedition if they don't let him go along with them, all go off in search of this mysterious spacecraft of Taylor's.  Their ape minds can't comprehend of there being more than one ship, and they don't really know that Brent didn't come with Taylor on the first trip.

Here is "The Mystery Photo!"   Thanks to Zachary Scott for providing it via snail mail.  This image was taken from a 'fan magazine' produced in the early to mid 1970's.  It may be a picture that was never used and ended up on the cutting room floor.. I don't remember this scene in any of the apes movies, correct me if I'm wrong.  Is this an unproduced scene from BTPOTA and does it show Zira, Cornelius (both easily identified) with their colleague, Dr. Milo?   Did the three chimpanzees head into the Forbidden Zone in search of Brent's ship after Brent and Nova left looking for Taylor?  I believe so.

After a few days searching, Cornelius, Zira, and Milo find Brent's ANSA spacecraft by looking in the logical places (probably saw a glint off the hull from a tall hilltop or one of them climbed up in a tree and looked around) and investigate this mysterious ship.   What a marvel!  Those design lines, and look, a ladder going up to a door in the ship!  Could it be?  Yes, the three chimpanzees allow their curiosity to get the better of them and then use their scientific principles and training to figure out how man or ape can possibly fly (i.e. poking and prodding the strange whirring and buzzing boxes and buttons).  It is with this aspect of the reasoning that I see that Milo is the one that causes the problems.  He seems to be a 'very matter of fact' ape in that he's a 'know it all'.  I suggest that all of his big headed bravado results in a bunch of button pushing at totally random.   Unfortunately, they find out the hard way that the spacecraft is automatically programmed to launch and return through the Hasslein Curve back to Earth. It is a rescue ship, after all, and it was built so that in the event that the crew is heavily wounded or incapacitated, that a few simple, easily marked buttons will initiate automatic launch sequences and the computers will return the spacecraft to Earth.

Why the ships were so close yet avoided Gorilla patrols is easy to explain. The Gorillas were preparing for war on the mutants in the Forbidden Zone, bringing in all of their resources in the process. This would mean that vast areas of the Forbidden Zone, as luck would have it, were cleared of gorilla patrols. Their still active patrols were limited to just near Ape City and in keeping an eye on the mutants. The ships were far enough out that the patrols that would have been in that area had been called back in closer to Ape City days or weeks earlier.

While Taylor and Brent and the House of Mendez are fighting the Gorillas in the last armed conflict this planet will ever see, our three chimpanzee scientists are ‘monkeying’ around with the controls and they press one button too many. The automatic sequence starts, sealing them inside the ship, and asking them politely to don their flight suits. Well, as the fully functional ANSA spacecraft of Brent blasts off for the Hasslein Curve in space, it is none too soon as Taylor throws the switch on the Doomsday Bomb and destroys the Earth. The third, fully functional ANSA spacecraft returns its three flight suit wearing chimpanzees back to Earth in 1973. During reentry, the second stage / landing stage of the ANSA spacecraft is ejected and what we see is the three seater reusable command module which returns to off the California coast.  Of course, the government and ANSA (now NASA in the second movie) have tracked the spacecraft and dispatch a naval emergency spacecraft rescue team to bring the spacecraft in (...why it is hauled up on the beach instead of being brought aboard an aircraft carrier is up for debate, but I guess the budget for the movie had some limitations and unless you are a Kennedy, you just don't have an aircraft carrier on call to look for you if you are missing or you go down in the drink).  So we have to improvise, and Brent's spacecraft is pulled up onto the beach by tow cables.

The first person on the scene, of course, quotes that it is ‘Taylor’s ship!’, but it isn’t.  All of these ANSA spacecraft are similar in appearance except for minor differences.  Seeing one from the air in a chopper as the spacecraft wallows in the ocean waves might be enough 'fudge factor' for an untrained observer to make the mistake in identification of which spacecraft has really returned.   Also, since I have mentioned that so much is riding (politically and financially) on Taylor's mission, naturally there will be a lot of enthusiasm for the mission and Taylor to succeed.  It wouldn't be too far fetched to assume that the mistake made in saying that the returning ship is 'Taylor's ship' is simply chalked up to over enthusiasm and the genuine desire for the ship to truly be Taylor's ship.  But sadly, it is not, nor can it be in any stretch of the imagination, Taylor's ship.  We all know that, and I think that the mood really kicks in when only three step out of the ship, and they know that not only is this spacecraft definitely not Taylor’s four seater ship, but after sending three very expensive ANSA spacecraft into deep interstellar space that no human astronauts have returned.  Some heads are going to roll and I bet the deniable blame parties start in full swing.

When it is stated that NASA has been over the spacecraft and it is stated that it is 'Taylor's Ship', I think there is also an explanation here.  A lot of people's jobs and careers are riding on Taylor's mission, so a little lie here to someone who probably wouldn't know the difference between Taylor's ship and Brent or Skipper's ship might just save enough time to come up with enough data to actually save their job / career or maybe is all the explanation that is needed.  Getting back 'Taylor's ship' might be enough to keep the space program alive, to present a 'successful failure', as it were, much like Apollo 13.  Also, since ANSA sent the spacecraft up, and NASA recovers it, there might not be full mission and data disclosure between the two agencies, so NASA, investigating a craft that is not their own, might honestly make a mistake and say that, according to their knowledge, it is Taylor's ship.  I think that the chronometer and the onboard computer would verify if it was or not.

What is meant when Brent's ship is (mistakenly) referred to as Taylor's ship is that it is the same type of spacecraft as Taylor left Earth in (which further suggests that there are other designs in operation!).  Kind of like the when the movie "Smokey and the Bandit" first came out, that every black and gold Trans-Am instantly became 'The Bandit' Trans-Am.  I think the same kind of media buzz could happen to a mission as important as Taylor's.  I mean, come on, really, who could tell the difference between Apollo 10 and Apollo 13, but there were subtle differences.   Taylor's ship could have been the first of the series, with the others waiting to go (thus the quick rescue launch time), thus every capsule after Taylor's, especially with all the fanfare of his mission and launch, would have been a 'Taylor's ship'.  The popular buzz might be that all 'other' ANSA spacecraft built on the "Icarus" type spaceframe would be 'like Taylor's ship' or considered to be 'Taylor's ship' in that it was 'like Taylor's ship' and not 'exactly Taylor's ship'.

Again, what we might be missing is some off camera conversation between the pilots of the helicopter that discovers the ship.

What is being implied here is that it is "Taylor's TYPE of ship" and not necessarily the exact ship that Taylor commanded, but rather one similar to the one that Taylor commanded.  It is pretty evident that this simply cannot be Taylor's ship because, as we all know, Taylor's ship not only sank, it lost all power, developed structural damage, had the escape hatch blown off, and did not have a hatch behind the front view port where the ship we see in the third movie did have such a hatch.   Looking at the wreckage of Skipper's spacecraft, we also do not see a hatch behind the view port, so this rules out the ship that the apes return in being either Taylor's (which sank) or Skipper's (which crashed and burned).  Since Skipper, rendered blind in the crash of his spacecraft, also sustains enough internal injuries to die shortly afterwards and Brent sustains no injuries whatsoever, we must assume that there is a third ship somewhere off camera.  That can be the only explanation why both astronauts would be sitting nearly side by side and one sustain mortal wounds and the other be unscathed.  There was a third ship, similar to Taylor's ship, and this ship was flown by Brent.  This third ANSA spacecraft is not only fully intact, but fully functional and probably highly automated which includes a navigational course set in for the trip back through the Hasslein Curve and to Earth.

Also, Brent says that he took a Earth Time Reading and got 3955 AD.  The ship that comes back has a chronometer onboard that reads 3955, verified by NASA and other agencies.  Since Taylor's clock read:

therefore, this ship cannot be Taylor's ship that returns, unless somehow the chronometer reset itself (for an interesting parody on that aspect, check out the ASTRONAUT FIELD MANUAL).  You would think that if Brent and Skipper were in the same spacecraft, then they must be sitting fairly close together.   You would think that if Brent looked over and took a Earth Time Reading, that he would have told his commander that they were lost in the future then instead of right before his skipper dies.

I think that there were two spacecraft for the simple reason that Brent was able to take a chronometer reading and Skipper didn't, which to me means that he was following Skipper into the atmosphere, slightly behind, with some time to kill before he actually had to take the controls of his own spacecraft.  Skipper also didn't have a clue as to what the chronometer said, and went into catatonic shock when he was told what year they were in.  This meant that he was obviously very busy with his own spacecraft, possibly with a malfunction that only got worse, maybe he did some loop de loops like Taylor's ship did in the first movie before he managed to get his ship on the ground (maybe it was on fire even as it landed).  You would think that sitting side by side, that he would be able to look over and see the chronometer from his seat but he didn't or couldn't, so that is how I take it to mean that there were two separate spacecraft, or if Brent did take a reading when he was sitting side by side with Skipper, right before reentry, that he would have found the fact that they were 2000 years in the future to be something worthy of telling his commander.  Since he did this reading prior to reentry, and he didn't tell his commander, I feel that they were not in the same spacecraft at all, but two different ones.

In "Escape," the presidential council mentioned two missing spaceships, not three. So? Why couldn't the council be talking about the 'LAST two missing ships', as in the two MOST RECENT missing ships, as in an expensive rescue mission mounted to rescue an expensive first mission in the first place?  They just sent out two spaceships (Brent and Skipper's) and they only get one back!  That's not very good, and the public is going to want to know why that happened.

Taylor's being the third, or first in reference to this depending on your point of view. The most recent mission, and the fact that one ship came back out of two (or three) sent, and it came back with three apes probably has the presidential council biting their nails wondering how they are going to explain that.