There are multiple reasons for the interest. But before I get into it, allow me to ask a rhetorical question:
Why is there interest in any race but Humans? Why bother with elves, dwarves, etc?
The answer is: because they're different than humans. In some fashion, at least, people choose to not play a human either for the mechanical benefits, or for the story benefits.
1) Same idea. It's just that animalmen are further from human than demi-humans. This is one reason why I like them; they're inhuman. A nice distance from the typical elf/dwarf/halfling.
Not only aret they far enough from humans/demi-humans to be interesting, but familiar enough to be rationable. We all understand animals, their behaviors/habits and appearances. So something that is partially animal is a little less alien to us than say, something that's part plant. We don't understand how a sentient plant would behave.
Because we can understand animals and their behavior, it's a great inspiration for characterization. I can tell you right now I can use a cat as inspiration for my catman's behavioral patterns/personality. From Aesop to Redwall, animal = personality traits, so it's a very easy shortcut. Also, taking those animal elements and implementing it into their culture is rather easy and interesting.
2) Variety. Aesthetically, animalmen are so very diverse. There's a lot of different visual forms out there. Way more diverse than the spectrum of humans.
3) Design inevitability. You can only make so many different humanoids that look like a human with long ears or green skin before they start all looking together. This is similar to #2, in that the animal kingdom offers more variety of different humanoids to use.
4) Mythology. There have been animalmen for centuries. Many of various gods have animal heads. The Minotaur. Faeries have animal parts (Satyrs and centaurs, particularly). Not to mention lycanthropes. Besides, it makes sense that a God with a Jackal Head might create Jackalmen in his own image.
5) ) Environmental appropriateness. It just makes sense for lizardmen to live in swamps, and live there comfortably. When you need to have inhabitants int he swamp, or something to encounter in the swamp, you look for what lives there and, like #7, give it weapons and intelligence.
6) Nostalgia. Many of the various animalmen have been around for a long time, and so some may have very fond memories of the first time they fought Bullywugs.
7) Everyone has a favorite animal. Or "If you could be an animal, what would you be?" So, it makes sense that someone who thinks snakes are badass might really like Yaun-ti.
8) Fear. We all have instinctive fears. The thought of a tiger leaping onto us and mauling us. Snakes underfoot. So being able to give those fears swords, and intelligence, allow them to be used to pose a better threat than just a wild animal.
9) Furries.