To be honest, I don't know why they're so popular.
The writing is not that good in terms of quality.
The female character seems incredibly dependent on the vampire, and she seems fairly weak. In terms of not being able to stand on her own, go after her own independence, her own happiness, etc. If that were my daughter, I think I'd have thought myself a failure as a parent. Independence, and being able to be happy/fulfilled with yourself and by yourself is kind of key to being able to share yourself in healthy ways with another person.
The while relationship is set on an odd note, i.e. she smells good to him the way raw steak smells to my dog.
Finally, when I read the books, it seemed like the author had taken A LOT from Anne Rice. In terms of how the vampires looked, acted, spoke, their morality, etc.
Obviously, I'm not a fan.

Oddly enough, I did like her other novel (this one for adults) - The Host. Although, it did seem like she stole the plot of Body Snatchers, she did put a unique twist onto it. Moreover, the relationships in that book were FAR healthier. And her writing was actually quite good in terms of quality.
I would have to say Twilight is popular because 1. teen girls often become obsessed/infatuated with people, and so the whole relationship speaks on their level, 2. the action is fast, so no one is bored with intellectual or philosophical discussions, 3. vampires have always been popular - from Bram Stocker to Anne Rice, 4. the reading comprehension needed for such a book is that of a grade schooler - so there's no strain in terms of reading.