Anthropomorphic animal characters. It's a multi-spectrum fandom.
The style spectrum goes from cartoon animals like Bugs Bunny to metaphorical works like "Animal Farm", Narnia, and "Maus" to hard science fiction characters like the uplifted animals in the books of David Brin, S. Andrew Swann, Andre Norton, Olaf Stapledon, and H.G. Wells.
The personal involvement spectrum goes from it being just another SF/fantasy genre to being a reason to hang out with friends on the weekend to being a lifestyle choice (philosophically or spiritually). The sex spectrum goes from XXX (which gets all the press) to PG (like the "Beauty and the Beast" TV show) to the "squeeky clean" G of cartoon fans.
The activity spectrum goes from couch potato to making fan art to making professional art to making movies. (You'll be blown away by Tim Albee's "Kaze: Ghost Warrior"; see http://www.kazeghostwarrior.com.) For costuming, most fans wear none at all, some wear only tails or ears, and some make impressive costumes for display in convention masquerades or for appearances at public events. The New England Fursuiters, a local group, is in big demand for charity events, store promotions, parades, etc.
The "being known in public" spectrum goes from embarrassment and hiding it from friends and work and family, to "who cares, it's just another hobby", to flaunting it and trying to provoke a confrontation from the mainstream (which gets all the press).
There are probably other spectrums I've overlooked, but the overlapping will get you people who are furry fans and don't realize it, some for whom it's just a hobby, and at the very extreme, those who modify their bodies with tattoos or surgery to resemble their "totem". (They, along with the "fursuit humpers", are the ones who get badly made TV specials and CSI episodes made about them that imply all furries are like that.) There are also Christian furries who use furry themes for parables and other religious stories and artwork.
That still doesn't explain Why.
For artists and writers, the appeal is that anthropomorphic characters can be used to explore humanity and social issues in ways that would be politically or religiously dangerous if human characters were used. Examples would be "Animal Farm" and "Maus" and "Pogo". They can also be used to explore what it's like to NOT be socially or physically human, to better understand the concepts.
For scholars, it's a modern mythology or folklore. The stories and artwork created by furries are comparable to the stories and traditions of various cultures that include anthropomorphic animals in their folk tales and myths. Reynard the Fox, Aesop's Fables, B'rer Rabbit, Coyote, and so on are all furry stories written before the fandom existed. One may make the case that because of them, there were furry fans throughout history who just hadn't found each other until the Internet made it easier.
For those in it for spiritual or psychological reasons, it's similar to animism or shamanism, or method acting, emulating animals with physical or personality traits one desires in hopes of their becoming second nature. Someone who wants to be more active and agile physically might adopt a cheetah character and act out enough to be that. Someone who wants to be more aggressive and confident might emulate a wolf or tiger. There's also the anonymity aspect of the fans being largely online or behind masks and being accepted for who they want to be and not who they're told to be or assumed to be. For some, the fandom is a gateway for them to really get out and meet people, make friends, and get a life, like social groups based on other special interests.
For those in it for sex, hey, it's kinky. You don't have to be a furry to like the foxy ladies and cat girls. (Or anime boi toys, if that's your thing.) But that's not what furry fandom is all about. If there are any fursuit orgies going on, they're going to be more like swinger's parties, and not out in the open or known about by most other furries. (In fact, I think that's what they actually would be, swingers' fancy dress parties with an animal theme.) You're definitely not going to have furries humping you in the elevators at conventions, like some nattering nabobs are claiming in forums like Fark. That would be plain stupid to expect or do in reality, and would get the doers thrown out very quickly and banned from future conventions.
After writing all that, I just realized I could have just said it's like horror fandom or anime fandom or any number of other fandoms, except with anthropomorphic animals as the focus. All that time I wasted.