What is backing USD or any currency other than the interest of the public? It's not like any country gets to dictate to other countries how much their money is worth.
More generally, currencies are backed by demand for that currency. A lot of that demand is ouroboros-style circular demand of two kinds: (a) shops demand the currency for their goods because their suppliers and employees demand it, who in turn demand it because shops demand it; and (b) people have loans because somebody else demanded currency, and now they will be subjected to demand for a longer time period.
Taxes are special in that they create an outside demand for the currency, so they act as the bootstrap and anchor of this whole demand cycle.
Also note that the sibling comment by jzwinck is confused: the Swiss central bank decided for some time to put an upper bound on the value of the Swiss franc. This is easy because they can always create more Swiss francs, and is irrelevant to your question.
This became news several days ago when they let their currency float again, resulting in a jump in its value and the ruin of multiple firms involved in foreign exchange trading.
