When someone "mines" for bitcoins, where does the money come from? I know back in the old days, when the U.S. printed money, they had gold to back it up. What is backing bitcoins? Could someone explain it in laymans terms?

Let's start with the first bitcoin. Suppose the bitcoin economy is worth 10 dollars and there is one coin in existence. That coin then has an inferred value of $10. Now let's say we want to verify that this coin is legitimate in an anonymous fashion. In order to do that we need to do some difficult calculations to ensure the coin is authentic. In order to finance this we create another bitcoin and give it to the person who does the work to solve the verification calculation. Now there are 2 bitcoins, but the economy is still worth $10 at the moment, so now the two bitcoins are each worth $5. Instead of paying directly for the service of verification, we paid indirectly through inflation.

So to answer your question, the money comes from the value of existing bitcoins, which are worth slightly less as we inflate (if we don't factor in btc exchange rate for a moment), but it is a situation that is well known ahead of time and accepted as part of the cost of owning bitcoins. When this process ends and there is no more inflation, transaction fees will pay for the verification since it is assumed that by then transaction fees will make up a bulk of the profits for miners anyway.