I'm answering your question assuming that you hope to be able to replicate it's own success in your own mobile app.

There are a couple of factors responsible for it's growth that are instructive to anyone building a mobile app.

"Leveraging the intimacy and privacy of the mobile phone."
We now have an *intimate* relationship with our phone like no other device in the history of technology. Every internet company that started before around 2010 has built their core interactions around "the old web" one which was accessed primarily via a browser on a computer.

Companies that start with a clean slate, should be building their interactions around how to do whatever the app is supposed to do while leveraging what is unique to people's relationship to their mobile devices. Photo-sharing has become a core part of the way we communicate now. Snapchat built something that provided an experience that leveraged the feeling of privacy and intimacy that is unique to mobile.

"Provided an escape from the "maturity" of other online services." Too many parents, aunts, uncles and other "old people" have encroached into the social networks of teens and young people. As a result, they've had a desire to find places to express themselves in places inaccessible by older generations. An important distinction is that it's not just parents and relatives that young people are trying to avoid, but also employers & colleges who are increasingly using "mature" social networks to review applicants.

"Leveraged PR even bad PR" The fact that the app got so much press about it being used to sext was perfect PR for the company, as it essentially reinforced the brand experience that it has today. Essentially, "if it's safe enough to send a sext, it's safe for any kind of communication I want to have." And although the safety and security of Snapchat is actually not as advertised, it still enjoys the reputation of having less impact than any primarily web-based service.

Building a successful mobile application is one of the hardest challenges to face designers, programmers and entrepreneurs in the history of writing software. Happy to talk to you if you're considering building a mobile app, about what I've learned about the "table stakes" for success.