Because it's a get-out-of-jail-free card. Kind of like in Monopoly.
The U.S. dollar is not backed by gold or silver, but you can use it to keep from getting jailed for failure to pay federal taxes.
It has value is because our courts say it does. And what good is the opinion of a court, which is, after all, just words on paper? Well, if you are condemned to years in jail by a court decision, you will, in fact, spend years in jail. So, yes, court opinions are words on paper, but they mean a lot more than greeting cards.
In the U.S., your income is taxed, and you need to pay the tax in dollars. Even if you earned your income through barter. For example, if you started the year with an old baseball worth $1.00 and, through clever trading, ended the year with a speedboat worth $25,000, you would need to pay income tax, in dollars, on the gain you enjoyed, $24,999. You would need to barter your boat for some dollars in order to pay it.
This is what distinguishes the U.S. dollar from Bitcoin. No one needs bitcoins. They are elegant and, in their way, beautiful, like a work of art. But you do not need them for anything. But Americans need dollars, if they want to stay out of jail. (Yes, I know, we need beauty to be happy. True, but not important here.)
This way of thinking about fiat currency is not a new argument. I do not remember where I encountered it first. It does not get as much "circulation" as it should.
UPDATE 3/14/14: Modern Monetary Theory emphasizes the role of tax obligations in the value of a fiat currency.
