This is actually a straightforward answer. Bitcoins get value because people want them for different reasons and they are scarce. Anytime that happens there is value. The amount of value depends on how much people want them and how scarce they are.
If you think about it the same applies to any commodity, like gold, for example. People want gold for different reasons, and it's very scarce. That gives it value.
The only question then is why would people want bitcoins?
There are many reasons, but some are obvious. Bitcoins are a store of value like any money, but any amount of bitcoins can be sent anywhere in the world, in seconds, for free. Think of the hurdles now sending money to someone overseas. There are all sorts of middlemen to go through before a recipient can get that money, and the process often involves fees and delays. With Bitcoin, however, the transfer is instant and can be free (or extremely low fee). This alone make bitcoins advantageous to use, especially for people conducting business internationally.
Another key reason bitcoins have value is they can be used anonymously. In modern times information on what we do and things we buy is often tracked by interested parties, like advertising agencies. What if you wanted to buy something of a private nature from some website? You probably would have to use a credit card. That means Visa, or whatever company you use, would have a record of the details of that transaction. Could you always trust Visa to keep all your information private? With Bitcoin you wouldn't have to worry. With Bitcoin you are your own bank. You have total control over your money and can choose to use it however you wish, without middlemen. Many people value that.
There are many other advantages Bitcoin has over traditional currencies, but those may be the most obvious. For confirmation that people value Bitcoin simply look to its price history. It has held good and increasing value since January 2009 when it first started.