Scarcity and money;
Money or gold or anything else has value due to the interaction of supply and demand. That is, there is human demand for a thing (for whatever reason), and there is only a limited, finite supply of that thing (it's scarce). VALUE arises from human interaction in society. It's a social thing.
When some resource is scarce -- and nearly everything is scarce, because that only means it's not readily available in infinite supply anytime anyone wants it -- when a resource is scarce, then the people who have or control it won't give it up unless they receive something desirable in return. And the people who want it must make a choice -- what other thing will they have to give up in trade for the desired object? Making a choice, giving up one desirable thing in order to obtain another desirable thing -- that defines value. VALUE is that quality of the scarce resource that makes you willing to give up something else in order to get it.
OF COURSE all value is arbitrary -- there is no such thing as "intrinsic value" -- it's all about human opinions and emotions and mass psychology and fashion, and willingness at any moment to swap one thing for another. Those opinions change, and so the value of things can change radically.
Money has value because people want it (obviously), and the authorities who control the supply of money ensure that it remains scarce. We don't produce infinite amounts of money, because we know if we did it would lose its value. Therefore money has value and clearly is not "worthless".
You can't deny there is demand for money can you? It doesn't matter WHY there is demand for it to have value. I don't understand why there is any demand for Justin Bieber music, but apparently there is, so it has value. But it's not hard to see why there is demand for money. By custom we use it to acquire things we desire. Our society has made a collective decision that little pieces of paper with a $1 printed on it will buy so much, and little bits of paper with a $5 on it will buy five times as much. We agree on that, and the monetary authorities strive to ensure 1's and 5's remain scarce so as to retain that value.
Currencies last because they work so well and there is a lot of inertia built into the status quo. We are paid in US dollars and use US dollars to buy things. How and why would that ever change? You may say money is worthless, but are you going to open a restaurant and NOT demand money for your food? If you opened up a store in Texas and demanded to be paid in Hungarian Forints, or tulips, or in-kind barter, you'd quickly go out of business. It's like natural selection -- approaches that don't work quickly die out.
Again, value arises from interaction in society. Society has set up guidelines such that money has a purpose and retains value according to basic rules of supply and demand.
More deeply, people demand money because it is an incredibly valuable tool in many ways. No we don't trade goods and service directly because you will seldom find that YOU have what someone else wants, when THEY have what you want. Economists calls this the "coincidence of wants problem". You can have a crude, inefficient, localized economy based on such barter -- like the ancient world did before money was invented 2600 years ago. But you don't get far past the stone age without money, and we'd certainly never achieve anything like modern civilization without money.
Furthermore, a money system and a pricing mechanism serve to transmit information about what is needed, when, where, how much, and by who. This is called the allocation problem. In any moneyless society you'd never be able to allocate resources efficiently so as to satisfy peoples' needs very well.