Is this due to a matter of constant paid advertisements online, or is it just due to a viral word of mouth?
I've been in the picture messaging space for a while now with my apps Lutebox (voted one of London's top ten most loved apps) and now Click Messenger. I've written a few articles about the space including a recent post about the Future of Mobile Messaging.
Snapchat started out as an app called Picaboo, which pretty much did what it does now (prior to the latest update with chat and video calling). They quickly rebranded but saw a little uptake in user numbers and had quite low downloads for several months. Then around Christmas 2011 one of the founders' mom had told her friend about the app, who told her kid and her kid basically then spread the word throughout their high school in L.A. That was what really blew up their download numbers as it spread across teenagers at local high schools.
As far as I know they didn't advertise in the early days, relied solely on word of mouth. Also it is assumed that they have a solid user base. Comparatively speaking, their user base may be in the low tens of millions, which may a great base of users, but nowhere even close to being as big as Facebook or Twitter.
I'd be happy to speak about this in more detail or about the picture messaging landscape and what I believe to be the future of mobile messaging.