Every American participating in that economy must pay taxes every year on their income. The taxes must be paid in US dollars, even if the economic activity uses a different currency or barter. Therefore Americans must come up with a quantity of USD proportional to the size of the American economy every year (or "go to prison"), and provided that the total amount of USD in existence is bounded (this part is the job of central banks) this places a floor on the value of USD. This story is necessarily simplified but I think essentially true.
There is an old saying that a language is a dialect with an army and a navy. Likewise, a fiat currency is a currency with a taxing authority.
Gold, on the other hand, is just valuable because it is expensive to produce and has a long history of being valuable. So that model can work.