We say it because it's fun, has a nice rhythm and (almost) rhymes. The full exchange is 'See you later, alligator!' 'In a while, crocodile!'

En verdad, yo creí que "see you later, alligator" fue un dicho de los años 1950 y 1960 y no había sobrevivido. I thought that "see you later, alligator" was a saying from the 1950s and 1960s and that it had not survived.

Hola Lupita. A mi entender es la forma en inglés de decir lo mismo que aquí con aquello de: "Hasta luego cocodrilo" ;-)

"See ya soon, baboon", "Next laugh, giraffe", or anything you can come up with. It's not meant to be an insult, it's just a silly game.