I'm not the typical demographic for Twilight Fans. I'm 35-44, male, married with 3 kids and I watch 6-10 hours of sports on TV every week
My wife got into the books through friends and described them to me a bit, but with the disclaimer that they were too "high school girlie" for me. I shrugged and took her word for it until we bought the Twilight DVD and watched it together, at which point I was like, "hey, that's a really good story." I finished all four books in five days, which did involve staying up until 3 am twice that week. At this point, I've read them all seven times, though did skip some chapters of New Moon for some of those reads (the depressing parts).
As for why, I've always enjoyed the whole vampire mythos thing, though never a fan of the horror/gore genre (i.e., Blade or Underworld is more my speed). But more than that, I found it a very compelling delivery of first person, which I know from experience is very difficult to write in a manner that both captures the stream of consciousness thoughts of the protagonist while still describing the scene and flow of events. Stephenie did a great job of delivering on both required aspects.
Oh, yeah, I'm also a bit of a hopeless romantic, which I'm sure didn't hurt
As a parent, I like the fact that the stories can be read by my kids without encouraging them to act contrary to the morals I've taught them.
As a former teenager, I appreciate the trip down memory lane, the high school dynamic, and how your first love so completely takes over your personal universe.
As an adult, I love the unique take on man's struggle to suppress the natural man, and the spiritual questions about the worth of one's soul and how much our choices influence our fate in that regard.