I do not know. I wish I did...
Maybe because I am a little geeky and just find it interesting how they work. And then it went from there to being something I never got to fully experience, instead just watching my classmates go through these complex treatments. When I was younger, the kids with bionators and frankels and etc got a lot of attention showing off their appliances. It wasn't until we were older that braces were seen as embarrassing or at least undesirable.
But the embarrassment factor is part of the attraction too. And not being in control of how the treatment goes. I like the idea of being "surprised" by the orthodontist, too, like, surprise, you are getting headgear today! I like how braces effect the speech, and I like non-braces related speech impediments too. Like someone who stutters would be fascinating and sexy to me, so I guess there is some crossover there. But I still have no idea why.