Why does the US dollar have value?
Same reason anything has value: supply and demand. The supply of dollars is essentially determined by the Fed, and demand is driven by its ability to buy things (think: why do you want money?). At the intersection of supply and demand, you have the "price" of a dollar, found just like the price of a bushel of corn or a neglected Beanie Baby.
So, how does a Bitcoin come to bear the price it does? Supply and demand again--except supply is controlled by an algorithm instead of the Fed, and demand is driven mostly by speculation and its utility in sending money without having it all stolen by PayPal. Increasingly, you can also buy things with it.