From Catsparella via Catster...

In 2001, with the help of Best Friends Catnippers and FixNation Disney began instituting a Trap, Neuter, Return policy to help manage the feral population in the park. The cats are caught and fixed before being released back onto the grounds, and kittens are all promptly adopted out. Cats that remain in the park receive food at five discrete feeding locations, as well as shelter, and in return they work as part of the nightly maintenance crew, keeping the non-Mickey mice in check.

Since the cats are feral, they shy away from human contact which is why they mostly remain hidden during operating hours.

Some of the cats even have their own favorite areas, such as the one above who was spotted outside at Hungry Bear Restaurant, and another named Lucifer who reportedly spends his time between the Haunted Mansion and Town Square.