US paper money USED TO BE redeemable for a specified weight of silver per dollar of face value. The exchange rate was set by law. In contrast, the "Federal Reserve Notes" now in use are not redeemable (by law) for anything other than US Treasury bills.

So, why does the USD have any value at all?

One reason is because the law gives them the status of "legal tender," which means exactly what it says on US paper currency: You have the right to use it to satisfy debt obligations, including taxes due.

Creditors MUST, by law, accept debt repayments in the form of US dollars. Refusal to do so results in the courts deeming the debt to have been paid anyway—but only if the repayment is in the form of cash or a cash-equivalent instrument (e.g, a bank account balance increase,) or is successfully converted into such by the creditor, such as by depositing a check that subsequently clears.

The other reason is simply because society chooses to accept payment in USD, at whatever rate of exchange the market will bear.