You and I. This includes the government, and a little bit of faith.
A “value” in the fiat world (IMHO) is an abstract term. It comes from demand and supply. What people are willing to assign for purposes of a medium of exchange.
In a world of very little items, for example, consider Corn, Rice, Chickens, and Wood as the only tradeable commodities in an economy. With four items over time, it will be easy to set the ‘exchange value’.
1 bag of rice for 3 chickens.
1 chicken for 2 small bags of corn, etc.
When you have 100s or 1,000s or 100,000s of items in a society to trade, near and far, then the medium of exchange becomes questionable, as it is not back by anything (used to be gold). Then you have to ask if it is not back by gold, then what is (fiat) money back by?
The answer is – it is backed by the issuer, again in a somewhat abstract sense called: The full faith and Credit of the Government.
We (the society) are ready to use this money and exchange our services for it, or our products, knowing full well, that this paper (or electronic) I.O.U. can be exchanged across the town, across the country and even across the world for other products and services.
As long as that faith exists, you have essentially assigned a value to the money, by backing the faith part of it.
When you lose faith in your money, it loses its value.