It's a pretty long answer. My colleague recently published a several page analysis of Snapchat's growth (link below). But a quick summary is that Snapchat is almost the perfect product for driving natural word of mouth and engagement. It was released by Stanford students for the iPhone, but didn't really grow quickly until it was discovered by high school students. Growth was amplified by the Android release and international expansion.

Here's the link to the full article:

http://www.growthhackers.com/companies/snapchat/