There are several reasons SnapChat was able to grow so quickly. Here's my take on the biggest factors.
1. Network Effects/Communication apps - the most viral products are those that have increasing value as more friends use them. Typical examples include FB, LinkedIn but even more extreme as pure viral products are communication tools such as a phone, text message or email, which effectively have zero value without at least 1 other person using them. Products such as these face a huge challenge of making something addictive enough that it will be used at high frequency between friends but if they're able to achieve this they can generate almost indefensible network effects.
2. Reduced friction - photo-sharing is one of the most frequent daily behaviors of smart phone users. By making photos private and disappearing Snapchat reduced the friction or hesitation to share since: a) you're more likely to share a goofy photo and send more per day if it's not on FB and b) you're more likely to send something a little edgy (whatever that means to you) if it's going to disappear. In addition the UI is one screen which makes it extremely easy and fast to send a photo or video in the moment
3) Stickiness - snapchat's disappearing photo feature makes each incoming snap a surprise. There are several games that have leveraged this technique for years to get users to come back but the brilliance of snapchat is that it's baked into the product itself
4) Fun - by building a product that takes several features users are already doing but packaging them together SnapChat has created a fun, addicting and unique brand. Being able to send a photo, draw on it and add text make it more addicting.
5) Mobile & app rankings - it's hard to get noticed or build a growth strategy around a mobile only product. The best chance of doing so is to build a great product that leverages word of mouth, has great design, low activation barriers (simple) and some form of virality built in (easy way to invite phone contacts or syndicate to social networks). Once you do however reach a wide enough audience, appearing at the top of the charts in app rankings adds additional fuel to your mobile growth.
For the past several years companies have leveraged FB as the most effective viral strategy but I believe we are seeing a shift towards your mobile phone & contacts as being the biggest untapped distribution platform and one that FB cannot control or police