Yep, this is one I've heard of before. I ended up watching a documentary about it one day because I was really confused by it all, but it's been going on for a lot longer than you might think. The correct term for it is autozoophilia.
I think animal play originated in American Indian tribes who used to partake in a lot of spiritual work, like getting into the mind and spirit of other people, and even animals once they had passed away to try and ensure they had an easy passing. Obviously, this might have nothing to do with it and might just be some sort of ritual, but this was the start of the documentary.
As I watched the documentary, I actually found myself understanding the appeal, well, the cat aspect anyway. The dog and pony was maybe too much for my liking. But with the cat play it was all about seductive feline movements, and being stroked and looked after, and it all just seemed really lovely and quite relaxing! I've never tried it though, I don't find it a turn on, but at that moment I kind of 'got' why people liked it. Some people just like the aspect of mentally being out of the human mindset and obviously there's the nature of someone taking care of you and all the affection that comes along with it.
There's totally different aspects of it, someone can play a kitten at home and just do it for a bit of fun, whereas others prefer to go into it. A it more. For example when they were talkinfn about pony play, some of the people would enjoy it beig fully dressed up, to the point of wearing masks, tails and even having their arms and legs bound together individually, so they were walking on their knees and elbows. Someone even made modified 'hooves' that had a lot of cushioning and padding to protect their knees and elbows meaning they could remain in the mental state for longer because they were more comfortable. To the point where some people chose to live like that 24/7 whereas others refered to themselves as 'hybrids' and it was a regular but a part time experience.
Personally I think it's a sort of 'expansion pack' of a Dom/sub relationship, most (but not all) Dom/sub relationships involve a colar and lead, and I believe this role play is just an expansion of that.
I agree with what AA said earlier, pony play tends to focus more on the strict training, and it a more advanced state of pet play. As Ness said earlier there are different types of play, like cart play, riding play and show pony play, but this can be applied to anything anyone does sexually, it's just about what individual people prefer. Puppy play is more like gentle training with playful aspects, but some of then go to the full extremes and the 'puppy' will have to use puppy pads, sleep in a cage, and eat from a bowl. Cat or kitten play is genrally just about the love, affection and just having a nice cuddle! There's also cow play, that hasn't been mentioned. This tends to be for people who have been into pet play, but become pregnant or want to become pregnant. It's focus is more on lactation and impregnation.
I did find a lot of different aspects when watching it though, for example there was a Dom/sub relationship and the female was the sub. She changed into a cat woman, but he kept referring to her as a stray and she was acting a bit wild and more dominant than she would if she was being submissive, so maybe for some people it's a break from being submissive? I don't really know, I think it really is just a matter of personal opinion and what suits the people involved.
Obviously, all of the above is fully consensual between players/partners, and safe words were always used. Jeeze I'm surprised I've remembered so much of this considering I watched it about 3/4 years ago. But I enjoy watching documentaries about different aspects of sex to just get a broader understanding of it, it's really interesting. This particular one was a part of a series which involved age play, gender play and general role play. It was pretty interesting!