Why does fiat money have value?
The answer, that can be found in greater detail in the following book

Rudolf Richter, Money — Lectures on the Basis of General Equilibrium Theory and the Economics of Institutions, Springer, 1989.

is quite simple. One has to create a positive marginal utility to the user. How one can achieve this aim? One has to create a positive demand for fiat money, which can be accomplished that taxes can only be paid with fiat money. Then it has a positive marginal utility, and the trick is done.