Fiat money is very confusing. It is magic, and to both stupid people who focus on it and smart people that don't it is too good to be true, so it must be crap. The thing is, though, that the gold standard is pretty much the same, except that it is hugely beneficial to people who managed to get some wealth together in the past. Gold's "intrinsic value" as a currency comes from the fact that it is a noble metal (doesn't go bad), it is relatively easy to verify, it can be broken into smaller units, and so for people with less technology available is an excellent source of liquidity currency. Unfortunately it is relatively fixed in supply, so it can't well accommodate growing economies (necessary deflation will always end up killing any growth potential).
Basically the only way for prices to be stable or (prefereably) inflate over time is for the amount of gold found to meet or exceed the growth in the economy. That may be possible over certain periods, but not in a predictable way or for the long run: the world has a fixed supply of gold, and after enough time the only profitable industry would be mining gold.
So gold is a pretty good currency 1000 years ago, but really, really bad today. What about the magic of fiat money? I think Brad DeLong has up an excellent overall post that gets down to why any advanced economy needs (yes needs) fiat currency.
It is simply not the case that we can cheaply and easily buy things with money because it is valuable. It is, instead, the case that money is valuable because we can cheaply and easily buy things with it.
One way into the tangle of understanding why it is wrong is to ask each of us: Why are you happy accepting money in exchange when we sell useful commodities?
Hint: It's not because we are looking forward to going down to the bank, exchanging our bank notes for the little disks of gold usually decorated with pictures of bearded men on one side and allegorical female figures on the other with lettering saying things like "Fecund Augustae" or "Concordia Militum" or "Fides Exercituum" on them, taking our little disks home, and feeling happy looking at them.
That's not why we accept money.
We accept money because if we don't have any money we have to buy commodities with other commodities, and when we do so we are unlikely to receive the cost of production for what we sell. Have you ever tried to buy a latte at Peets with a copy of Ludwig von Mises's Money and Credit? It does not go well.
The fact is that your wealth is only worth its cost of production if you are liquid--if you can wait to sell until somebody willing to pay full cost of production comes along, which is not every minute. The use-value of money is that it allows you to time your other transactions so that you can realize the full exchange value of what you sell, rather than having to sell it at a discount.
I don't think there is a good way to summarize, but the way I try and think about fiat money is this: the worth of an economy is measured by its total production of goods and services, not the amount of gold it possesses. But direct trading of goods and services is difficult so something liquid should exist to make transactions easier. That something must, however, be able to grow (and shrink) to accommodate the changes in the total economic output as well as to provide stability with some modest inflation (deflation is bad, bad, bad). Gold cannot do this, but fiat money can. If you want an economy that grows, innovates, advances, and improves the lives of everyone, you need to have fiat money flowing through that economy.
Aside: How much inflation? Historically it looks like ~4% is probably good, but I like the arguments for using NGDP (~5% total inflation plus growth) which is probably a better target since it allows for inflation and growth to trade off without running away.