The most immediate reason the play is called "The Zoo Story" is that Jerry announces that he's been to the zoo when he first shows up. His story is a zoo story, and that starts a lot of the play's action in motion.
There are other less obvious /more symbolic reasons as well. Jerry lost his parents; many animals in zoos (especially older zoos) are without full family structures, and Jerry fights for territory like an animal. The characters in the play are trapped and lonely, like zoo animals. Animals play a part in discussions.