Clothing has alot of purpose besides just covering one's skin. After all, in mostmany places it is now acceptable for men to be shirtless, and in some for women to be shirtless, and yet the vast majority still wear their shirts all the time.
Clothes help protect us from minor injuries like scratches and scrapes, as well as from the heat or the cold. It's true that these animal people would likely have fur (or at least many of them, idk if you're including reptiles or furless mammals), but even those with fur such as dogs still have places which can be rather sensitive such as the lower belly, the genitals, and the paws. Gloves/shoes and shorts would be very protective clothing.
Clothes are also an excellent way to wear your belongings which otherwise would have to be carried. Your wallet, a favorite pen, an ID, a handy tool, a weapon, and more can be carried in pockets or bags, built-in or attachable (or wearable as a separate piece, like a purse). Clothing is sometimes used to include jewelry as well, such as necklaces.
Lastly, clothes can be a great way to really trim out your look. Letting your hair grow wild can be an attractive option to some, but some people keep themselves trimmed up, and some like to wear specific clothes simply for how it changes their look. Animal people, to me, would essentially behave like people but look like animals; therefore, they would care very much about how they appear to others. If the right gloves could give you a look of superiority or finesse, I can't imagine some of them not wanting to wear them.
(Edited "most" to "many" in the first paragraph for better accuracy)