The preceding film, 'Beneath the Planet of the Apes', ends with a nuclear weapon's destruction of the apes' future Earth itself and everything on it.
'Escape from the Planet of the Apes' begins by establishing that three chimpanzee apes, Cornelius (Roddy McDowall), Zira (Kim Hunter), and Dr. Milo (Sal Mineo) escaped the Earth's destruction by salvaging and repairing the astronaut Taylor's spaceship (which sank in the Dead Lake in the desert Forbidden Zone in the first movie) and piloting it through the nuclear shock wave of Earth's destruction, sending the ship into space and through a time warp wormhole back to present day 1973. The salvage, repair and launch of Taylor's spaceship apparently all happened (off-camera) within the brief period during the final act of the previous film.
The three apes arrive on Earth, splashing down on the Pacific coast near Los Angeles. They are found by the surprised local authorities and are transported to a secluded area of the Los Angeles Zoo, under the observation of two scientists, Stephanie Branton (Natalie Trundy) and Lewis Dixon (Bradford Dillman), who administer a series of intelligence tests, which Zira performs unerringly, but she finally loses her patience at the end of the last one and answers Stephanie's question to Lewis with an angry retort. Stephanie and Lewis step outside, completely astonished. Milo argues with Zira and Cornelius about their agreement to hide their ability to speak until they felt the time was right. The argument agitates a wild gorilla on the opposite side of a set of bars in the cages. Milo steps too close to these bars and the wild gorilla reaches through them and strangles Milo to death.
Meanwhile, a Presidential Commission has been formed to investigate the return of Taylor's spaceship which was launched the previous year and how the apes, which they already are aware are atypically intelligent, came to be aboard it. Cornelius and Zira are brought before the Presidential Commission, where they publicly reveal their ability to speak, and are welcomed as guests. They tell the commission about their planet in the future which is future Earth, about the apes way of life, but they deny knowing Taylor or anyone else that crash landed on their world.
Over the next several weeks, the two apes become celebrities, being lavished with presents and media attention as well as staying in a fancy hotel suite. They come to the attention of the President's Science Advisor Dr. Otto Hasslein (Eric Braeden), who discovers Zira is pregnant and fears for the future of the human race after Zira had earlier told the commission about the growing dominance of apes to be set in the near future. Determined to force the issue, he gets her drunk on champagne (which he assures her is harmless, calling it "grape juice plus"). The resulting interrogation enables him to convince the Commission that Cornelius and Zira must be subjected to more rigorous questioning.
Both chimpanzees are brought to a top secret military base are questioned using various means of interrogation; during this time, one of Hasslein's assistants refers to the apes as "monkeys", stirring Cornelius' anger. Hasslein defuses it, saying they simply want to know how apes rose in dominance over men. Under interrogation, Cornelius reveals that according to the historic archive books and sacred scrolls in their ape society of the future, the human race will eventually meet its downfall in the late 20th Century after a plague wiped out all domestic cats and dogs of humans and that humanity adopted apes as household pets which they turned into slaves and that a militant ape, known only as 'Aldo', led a slave revolt and the world was soon dominated by the simians, which will, many years later, lead to the surface of Earth's destruction through nuclear war and anarchy. So many centuries later, in the year 3955, the apes have established a stable and peace full society of semi-primitive nature. However, there are still suspicions about how humans are treated by the future apes when Cornelius and Zira continue to deny knowing Taylor or talk about how humans are treated in their world.
Suspicion had already been aroused by Zira letting it slip, during her drunken interrogation by Dr. Hasslein, that she had dissected humans in the course of her scientific work. Hasslein orders Lewis to administer a truth serum to her, while Cornelius is taken to confinement quarters. Lewis warns Zira that the serum will have the same effect as the champagne Hasslein convinced her to drink earlier. As a result of the serum, Hasslein learns for himself that Zira examined and operated on humans in the future and indeed knew Taylor as well as Barnes who also arrived on their planet.
Zira is taken to join Cornelius in confinement while the paranoid Hasslein takes his findings to the U.S. President (William Windom) who insists to the President that the apes are a danger to the human race. A little later, an orderly bringing food refers to the unborn child as a "little monkey"; Cornelius has heard enough of the epithet and knocks the tray out of the orderly's hands and bashes him on the head with the tray, and uses this advantage to escape from the military base. At first, Cornelius thinks that he has only knocked the orderly unconscious, but he is later stunned to overhear that the man died. Hasslein uses the orderly's death and of Cornelius and Zira's escape as an illustration of the future danger the apes present and calls for the apes' execution. The President reluctantly orders that the unborn child's birth be terminated and that both chimpanzees be sterilized or killed.
Running for their lives, Cornelius and Zira (assisted by Stephanie and Lewis) find shelter in a circus run by a Mexican national named Señor Armando (Ricardo Montalbán), where an ape named Heloise has just had a baby. There Zira gives birth to a son, whom she names Milo (later known as Caesar).
Hasslein, knowing Zira will imminently give birth, orders a search of all circuses and zoos. As a result, Armando must send the apes away. Armando, Lewis and Stephanie provide them with maps and help them plan to hide until such time as they can escape into the Florida Everglades to live the rest of their lives in peace. In case they are discovered, Lewis gives Cornelius a pistol to use as a last resort. Zira asks to say goodbye to Heloise before they leave.
Hasslein tracks the apes down to an abandoned ship in Long Beach harbor, and finds Zira resting with her infant. After a short chase through the ship and to the docked ship's stern, Hasslein shoots Zira in cold blood after she refuses to hand over her infant and then proceeds to fire several shots into the infant. Hasslein is immediately shot to death by Cornelius, and falls overboard. After Cornelius kills Hasslein, he is shot by several Marine Corps snipers and, as Stephanie and Lewis watch helplessly from the shore, falls to his death onto the poop deck of the ship. The mortally wounded Zira tosses her dead baby over the side of the ship before crawling to lie with her husband and dying by his side.
But in the final scene in a surprising plot twist, Stephanie, Lewis and the other humans are unaware of the real fate of the infant ape. Earlier, Cornelius, Zira, and Armando switched babies with Heloise before their escape from the circus. Armando now watches over the infant Milo as the circus is being moved to a new location. Armando remarks how intelligent that the baby chimp is who will now survive with him at his circus. The final shot shows the baby ape Milo sitting in a cage, plaintively speaking the words "Mama? Mama?" with the voice of a human child.