Bitcoins have value because they are useful as currency. It has the characteristics of money (durability, portability, fungibility, scarcity, severability and recognition) based on mathematical properties instead of trusting in physical properties (like gold and silver) or trusting in centralist authorities (like the fiat currencies).
Summarizing, Bitcoin is supported by mathematics and cryptography. With these attributes, everything this class of money needs to maintain its value is to be trusted and adopted.
Concerning Bitcoin, we can measure it with its growth in users, traders and start-ups.
As with any currency, the value of the Bitcoin is obtained only and directly from the people that wants to accept it as payment.