Simply put, Bitcoins hold value because they can be used as money. Just like any other currency, Bitcoin value is greatly influenced by who uses the currency, how many users are using the currency, and how much of the specific currency is in circulation.
But unlike traditional fiat currencies, Bitcoins are not susceptible to the value of gold or silver, or authorities who decide how much money to print. Bitcoins are a product of pure mathematics and raw algorithmic calculations, and are only influenced by the amount of trust that its users put into the currency and how well it adapts to being used worldwide.
The merchants, business startups, and users determine the value of Bitcoin by choosing to use Bitcoin over other currencies. Simply put, the more people who choose to use Bitcoin as a form of payment, the greater the value of each Bitcoin will be.