According to this Forbes article, Snapchat started gaining popularity in Los Angeles high schools in 2011. This timing was important because Apple came out with their front facing cameras around the same time, so as the "selfie" became popular, so did the app. It's popularity grew as students encouraged their peers to download the app, similar to how Facebook became viral. The biggest key to the app's amazing success was the early team; they were incredible at listening to and understanding users' concerns and they made changes when they felt they were ready to do so.
Finally, Snapchat started making a lot of money when they provided advertisers and other companies with a way to individually target users in a way that no other app had before. It didn't feel invasive because you still had your personal side of the app, that included conversations with your friends and no ads. On the public side of the app, you could view your friends public stories with ads that you can click through, as well as stories from bigger companies like BuzzFeed and Cosmo. The bigger companies stories include ads that you can't just click through so you have to view the ad for a few seconds if you wanted to continue the story.
They have really turned the app into a viral sensation that is part of my everyday life.