To disagree with mmakunas, I don't think people care about the privacy angle so much- I think it's about creating something entirely disposable, and the freedom that gives you. It's like an anti-Instagram- no filters and perfectly positioned shots, just point, shoot, send.

I'm not sure what the exactly psychology of it is or whether it even matters that the photos disappear but a lot of teens use Snapchat. Just take a look at the demographics of a Twitter search:

https://twitter.com/search?q=snapchat%20me&src=typd

Even if Snapchat doesn't do anything that normal messaging can't, they've clearly packaged it in such a way that it's tremendously popular. After all, if it was possible with normal messaging, why wasn't anyone doing it until Snapchat came about?