Disclosure: I haven't seen this movie in yonks so I don't remember it very well. I am, however, a massive time travel nut. Here is the sentence which makes everything in the entire movie make perfect sense:

Time on Earth and time on the Planet of the Apes run in opposite directions.

Pericles the ape leaves the ship Oberon first and travels through the storm. Leo follows Pericles through the storm. Eventually the Oberon follows Leo.

Because time on the other side of the storm is running in the opposite direction, the three travellers arrive in the opposite order to that in which they set out.

The Oberon arrives first and crashes. Its ape cargo swarm out and populate the planet, creating the Planet Of The Apes. Thousands of years pass. Leo arrives next, eventually locates the ruins of the Oberon. During the climactic battle scene of the movie, Pericles finally arrives too.

Now we have everybody on the far side of the storm, and we move into hypothesis.

Leo returns to Earth in his pod, through the storm. Hypothetically, some time later, the apes develop space travel and follow Leo through the storm.

Because time runs in opposite directions, again the travellers arrive in opposite order:

The space-travelling apes land hundreds or thousands of years before Leo: in fact, hundreds or thousands of years in Earth's past. They conquer Earth and it becomes a new ape planet. HISTORY CHANGES. Leo arrives much later (but still in what is technically his own past because the movie starts out some time in the future). He discovers a regular Earth but it is now populated by apes.

Leo is back on Earth, but history has been changed and he has no way home.

Regardless of how good the rest of the movie is, this is a pretty cool and sophisticated model of time travel, and the final reveal makes complete sense, as well as being a mind-boggler in the best spirit of the original. I am led to believe that nobody involved in its production - even the director - actually understood the twist ending. I believe the only person who actually "got it" was the original script writer, and the twist was simply left in by everybody who looked at the script afterwards, each reader reasoning that the twist was still good even if they, personally, didn't understand it.

As a side note, there is no reason why the Planet Of The Apes can't still be Earth-in-the-distant-future, as it is in the original flick. In fact, this makes a great deal of sense: it would mean the storm simply connects two different periods in time rather than two distinct solar systems which both happen to have Earthlike planets.