[lottin]

> I'm confused. Isn't the US dollars something like a legally-backed "we-owe-you" of US goods and services?

Not at all. You can't take your dollars to the issuer (the Federal Reserve) and have them redeemed for goods and services. They are not redeemable, hence not "backed". Fiat simply means not backed.

> Why else do non-US residents value the USD?

Do you mean why they demand USD? For different reasons. One reason is to pay for goods and services imported from the US. Another reason is to use it as reserve currency.

> And why does EURUSD vary with relative economic output of the two countries?

It doesn't.