First, as others have pointed out, there's a promise of profits that SpaceX can sell. Similar to Amazon, who were and are selling investors on the promise of growing and owning online retail, SpaceX can sell investors on the promise of growing and owning the space industry--travel, mining, government contracts, and more.
Secondly, it's worth noting that "billions" isn't as big of a number as you'd think in the context of modern capital markets. PwC published a paper recently that estimated there'd be $100 trillion in so-called assets under management (AuM) by 2020. (http://www.pwc.com/us/en/press-releases/2014/pwc-global-asse...) The market capitalization of Amazon _today_ is $400 billion.
Theoretically, SpaceX could easily raise all the money it needs and more. I don't know how practical that would be, or even if that's their plan at all. But the salient point here is that there's a _ridiculous_ amount of global capital up for grabs. This ain't the 1950s or 1980s. The rise of the third-world and the pace of global trade has created an almost unfathomable amount of wealth. Even if the markets crash tomorrow and it's cut in half, it's still unfathomable from the perspective of 20 or 30 years ago.
I remember reading an Economist article in the late 1990s (maybe early oughts) that showed using historical data that wealth tended to grow exponentially. (I haven't been able to track it down.) The point of the article was that wealth inequality will naturally increase dramatically because for any time span those with more wealth will accumulate even more wealth, proportionally, then someone poorer. The gap will naturally widen without forced redistribution.
Ignore issues about wealth inequality for a moment and think about what that phenomenon means for the amount of global capital available for investment. All that capital needs to be _invested_ somewhere. Even if appetite for risk never changes, it means the amount of capital available for the most risky ventures will grow dramatically. But perhaps as capital growth outpaces investment opportunities, appetite for risk will increase.
My point being, the practicality of a privately-funded Mars venture is likely to become more practical by the year. So even if it's starry-eyed today, it might not be in 5 or 10 years.