What information is sufficient for describing a thermodynamic system? For a single-component system, why are the energy, volume, and number of particles sufficient for describing the thermodynamics of the system? Why just three variables and those three variables in particular?
In the book that I am using (Callen, $\textit{Thermodynamics and an Introduction to Thermostatistics}$) he postulates that the macroscopic equilibrium state is characterized by the energy, volume, and particle numbers of its components, but what is the reason for this?
 A: Typically, the state variables used to  identify a macrostate are N, V, E for an isolated system, or you can replace E with T if the system is in contact with an heat bath. Other choices  are also possible  (the different formulations are related by the Legendre transform).
Note that other substances may require an enlarged set of state variables (for example, magnets or superfluids, see https://arxiv.org/abs/cond-mat/0405111 ).
Generally, apart from V (that sets the "size") and the internal energy E (that is related to the microscopic Hamiltonian), the other variables should be "conserved quantities" under the (possibly dissipative) dynamics of the system: for a simple system N is a conserved quantity (it is a "Noether charge"), other systems (like superfluids) may have extra conserved quantities.
A: A flippant, but not entirely inaccurate, answer is that those three variables were chosen because it was convenient.
From the Wikipedia page for Thermodynamic State:

A thermodynamic system is a macroscopic object, the microscopic
details of which are not explicitly considered in its thermodynamic
description. The number of state variables required to specify the
thermodynamic state depends on the system, and is not always known in
advance of experiment; it is usually found from experimental evidence.
The number is always two or more; usually it is not more than some
dozen. Though the number of state variables is fixed by experiment,
there remains choice of which of them to use for a particular
convenient description; a given thermodynamic system may be
alternatively identified by several different choices of the set of
state variables.

