Visible crew/equipment: When the astronauts are running down a dusty hill in the desert, you can spot the shadow of the camera on the back of Landon and Dodge. (00:15:50)
Visible crew/equipment: Right before Taylor gives the female ape (Zira) the piece of paper with his name on it, while he is fighting the other ape, the shadow of the camera is seen on the bottom of the cage. (00:47:05)
Visible crew/equipment: After Taylor flees the market, he enters a museum and begins to run down a spiral staircase. As he reaches the bottom of the staircase, to the right of shot, a cameraman's shadow is visible on one of the carvings and the staircase's support column. (00:55:30)
Visible crew/equipment: After Taylor exits the museum he is whipped by an ape on a horse. Taylor/Charlton Heston is replaced by a noticeable stunt. (00:58:57)
Visible crew/equipment: When Zira locks Nova in Taylor's cell watch the left iron bar and you'll see the shadow of the camera approaching.
Answer: I think that this is meant to be a mystery. Taylor/Charlton Heston, an astronaut, leaves a world set somewhat in the future after 1968 (when the movie was made) but still recognisable to cinema-goers at the time, to travel through a "time vortex" to arrive in a world in a distant future, which has changed beyond recognition. Taylor meets the orangutan Zaius/Maurice Evans, and Zaius hints that he has some idea of what had happened, but Zaius' knowledge is either limited, or else Zaius is not going to tell Taylor (or his fellow apes) the full story. At the end of the movie Taylor discovers that, at some point between his leaving his own time and arriving in the "Planet Of The Apes", the world had been devastated by a nuclear war, but I think that the exact time, causes of, and course of this nuclear war are deliberately left as a mystery. Sometimes I think a bit of unresolved mystery actually improves a story, and I think this is the case here.
Rob Halliday