A valuation isn't just based on how much money a company makes right now. It's based on how much money the company's expected to make over its lifetime. Snapchat's valuation [1] is based on expectations of high future revenue.
(There was a point where Facebook & Twitter had 0 revenue, too. Now they're both multi-billion dollar public companies.)
These stories always cause scepticism, especially in the HN crowd. It's the most common response ("What?! 4 BILLION!?"), and an easy one (i.e. doesn't require much thought), but it's the wrong one.
Sure, you can argue over the exact number, but let's consider the reverse claim: are you really willing to bet that an app that processes 350m+ messages every day is worth nothing? If not, what would you value it at?
Answering that question requires hard thought; reflexive scepticism doesn't. And if you were really imaginative & thought it through, you'd probably arrive at quite a high number.
For many consumer apps, the path to market dominance involves offering a service for free at first, and then 'monetizing' later on. The whole point of venture capital is to allow that (common) pattern to happen; the result is massive companies - like Dropbox and Airbnb - that weren't always cashflow positive, and that meaningfully benefit the world.
Perhaps it was crazy of Snapchat to turn down $3 billion, and maybe they really are overvalued, but it seems obvious that they are worth a lot of money [2].
[1] Although valuations make great media stories, valuation is really not that meaningful a number for private VC-funded companies.
[2] If you don't agree, consider that both Mark Zuckerberg & the founders of Snapchat disagree with you. So either you're smarter than Mark Zuckerberg & the founders of Snapchat, or you're probably wrong.