Executive Meddling: The film was initially two specials set to air on NBC during the 1980 Winter Olympics and the 1980 Summer Olympics. Well, they managed to do the Winter Olympics just fine — but then the US boycotted the Olympics and Moscow, and the producers got screwed.
This was Steven Lisberger's first major project, and he used it to raise money to pitch for a certain film called TRON. The backlighting effects in Animalympics were the basis for the visual effects in Tron. Many of those who worked on Animalympics would also work on Tron.
You'll also see a lot of current Pixar employees here, too.
And Roger Allers, one of the directors of The Lion King (appropriately, he animated a lion).
In addition (among many parodies of then current sports announcers and celebrities) Billy Crystal did his Howard Cosell and Gilda Radner did her Barbara Walters (i.e. her Baba Wawa from Saturday Night Live.)