Frankly, it is just someone interested (fascinated might be a better way of putting it) in "anthropomorphism" - the idea of, essentially, a cross of human and animal characteristics. And it seems to mean pretty different things for everyone.
From the website for the largest "furcon" in the world -Anthrocon-
If you as an adult still occasionally like to flip to the old cartoons, or have a stuffed animal sitting on the dashboard of your car, or buy cereal because it has a cool tiger on the box, you may well enjoy what our fandom has to offer.
Just what IS "Furry" fandom? | Anthrocon 2011: The Anthropomorphic Institute of Magic, June 23 to 26, 2011
Sure, there are people in it for sex ("yiff" as was said above) but many are not. Like all "fandoms" (looks at StarTrek, StarWars, anime fandoms, etc. etc. etc) there is an element that is involved in sexual terms - pretty much seems to be a human things overall, sex is prevalent anywhere possible essentially.
But others are simply in it for enjoying the art, literature, and such things available to them from the fandom.
Not all that many actually even have a "fursuit" (which is what CSi and the like have relied on and used for the shock value - sex dressed up like an animal). They are very expensive and few people make them. And a large portion of those who do make them do not accept "adult" requests for the suits.
The last in-fandom study that was done (which is done pretty much yearly, I think):
Furry Survey Results: Current as of April 7, 2011
Sorry for the long post, maybe that will at least give people something outside the CSI episode to look at it by.
[Yes, I am a furry, hence the name.]