The way it was explained in the original series was....after the apes revolted and fled from the majority of humans, there was a nuclear war that killed off most of humanity and the apes slowly gained control.
I don't know how they will do it for this movie though. They have never been real consistant with the storyline. The 5th movie, BATTLE, suggested that the actions in it changed time and the events that Heston crashed into in the first movie never happened. In the 3rd movie, ESCAPE, they say that a plague killed off all dogs and cats which lead to the humans bringing in apes for pets which started the problem....but in the TV series when the astronauts crash in the future, the first ape that finds them is out playing with his dog at the time.