It has value because:
- Just like FIAT, we believe it has a certain value
- Because you can sell it at any time for that value
- Because it takes X amount of dollars worth of energy to produce a single Bitcoin (this is often overlooked - it established base cost)
- There is a limited amount of Bitcoin (whereas more new dollars get printed every day = inflation)
- It's better than old money, because you can send/receive it anytime anywhere and don't even need to be there physically
- By cutting out the middlemen (banks) you gain: Freedom of spending, private access to your funds (even police can't touch it without your passphrase), lower costs, faster speeds, a certain degree of anonymity/pseudonymity
And that's just at the current stage. There are many improvements and different implementations being built.
People ask me: "What real world value does a Bitcoin have?"
I ask them: "What digital value does gold have?"
You can't buy anything on the internet with gold. But you can with cryptocurrency. In fact, I am pretty sure soon gold will be tokenized into a cryptocurrency too. The true value of Bitcoin is that it can be used to buy digital things: parts of an infrastructure or digital fuel (Ethereum), or part of an online business (OmiseGo) or part of a new generation Facebook (Steem/it), and reap the rewards from that. Anybody who wants a piece, will need to have cryptocurrency (aka Bitcoin) to purchase them.
Gold isn't better because it's physical. In this new world it's worse because it's not digital.
Most of the gold in the world is still owned and monopolised by governments and central banks. Look at what happened to Qaddafi. LOL