Originally, Bitcoin had a value of zero. But then, a while back someone bought 2 pizzas for 10,000 BTC. This put the value of 2 pizzas into the overall system. It was a tiny amount but it still counted. Others have bought drugs and other contraband with BTC. Some have exchanged work for BTC. Some people bought electronics. All of these tiny amounts add up until a currency that is worth essentially zero begins to have value. A while back, a single BTC only traded for a penny but the "intrinsic" value was in there and it grew.
Others began to see this value and believed the value would continue to go up and, of course, they wanted to speculate on it and make some money. This speculation caused the price to rise as more money was pumped into the system. The intrinsic value hasn't risen nearly as much as the price, due to the speculation. However, the intrinsic value has risen because Bitcoin has gotten more visibility with all of the added speculation. This increased visibility increased the public's awareness of Bitcoin. It caused entrepreneurs and inventors to come up with new ways and new opportunities for people to use the currency. This cycle brings in more inventors, more speculators, and more end users. The speculators, unfortunately, drive the price up and are ultimately responsible for the peaks and valleys. But this isn't such a bad thing because every time the price rises, it creates more awareness and so the intrinsic value continues to slowly rise, even though it looks like the price may be falling today.
Ultimately, Bitcoin is "backed" by its usefulness. As long as you can do things with it, then it has value. The more people can do with it, the more value it has.
Consider this -
What if you could purchase something with complete anonymity? You can already do this with cash but can you do it almost instantaneously across the world with virtually no cost? This feature may not have value to you but it has value to others.
What if you could start accepting payments online and not have to: pay merchant account fees, worry about chargebacks, wait to get paid by the merchant, or have to set up a secure website? These features may not have value to you but they have value to others.
What if you did not like putting money into a bank, for whatever reason (maybe you're afraid the banks will take your money the way they did in Cyprus), and wanted to control your own money but still needed a way to pay others without sending checks, moneygrams, wire transfers, etc.? This feature may not have value to you but it has value to others.
What if you had a factory and wanted to export your products to another country but did not want to wait up to 8 weeks to get paid and purchase letters of credit for upwards of $100,000 each? This feature may not have value to you but it has value to others.
What if you were a world Superpower and were tired of purchasing all of the United States' debt and wanted a new currency to use as your "gold standard"? This feature has a lot of appeal to most countries in the world.
This is just a small sample of the usefulness of Bitcoin. Not everyone is taking advantage of the things Bitcoin can do for them today. Not yet anyway. But the list of things Bitcoin is useful for is growing rapidly and the number of people figuring out how Bitcoin personally benefits them is growing daily.