There's a lot to it, and it's likely some different mix of these and then some for everyone.

First, you got Disney and other assorted cartoons/games with anthropomorphic animals in them, which could possibly change your wiring a little if you're really into them while you're developing and forming tastes and preferences. Some easy (and commonly blamed) examples would be the Lion King, Lola Bunny and Robin Hood (the one where he's a fox). Hard to say whether this is a cause, an effect or both though, really. If you already tend to be attracted to animal people, you might latch on to one of these as a kid regardless.

Then there's the "confusing cute with sexy" thing. People think animals are cute/aesthetically pleasing/fuzzy and cuddly/whatever; you take that and throw in some parts that humans are wired to find attractive, and it easily becomes appealing. I suspect you'd find a similar effect in a lot of places, like with giant anime eyes and pet play and the like. Honestly, I don't think there's confusion so much as cute and sexy being very close to each other in our brains.

There's also the escapism aspect, which a couple people have already mentioned, and I'd say is mostly non-sexual. It's nice to imagine being stronger, faster, more fierce and wild or cute and cuddly when you're a strange, helpless, hairless ape thing - especially if you're a particularly socially awkward one, which many furries seem to be (not all, I know). Something that I think kinda ties into this is just the community aspect. When you're kinda weird, being part of a group of accepting, similar people feels good.

And to tie something else into that "stronger, faster, more fierce and wild" deal, I guess there's just something about being animalistic and primal that's appealing. That's a chunk of it to me, frankly. Consider biting/scratching type fetishes, leash/collar/pet play, the "in heat/breeding" type vocabulary you'll see sometimes, and even vore, etc. There are a lot of fetishes furry stuff branches into that can also easily be found independently, plenty of which you could characterize as "animal-like". Makes me think that's a common root, similar to fear and domination seeming to be the roots of tons of fetishes.

Zoophilia is kind of a different beast entirely, but likely shares some of this, sometimes with a little of that humiliation/degradation that would come with getting boned by a dog.

I think it's all pretty interesting and a lot more complex than it first appears. For me, though? I played Star Fox on the SNES, watched some Disney movies and always liked animals (no, I'm not sexually attracted to non-anthropomorphized ones). Guess that boiled down to always picking beast races in RPGs and slamming the ham to cartoon manimal porn. Nothing IRL with fursuits or anything, though. I know it's weird, but eh, everyone's into something weird. Makes it easy for me not to judge.