Why is it that the algebraic structure known as the Quantum Double $D(G)$ of a discrete group is said to classify the excitations of a Discrete Gauge Theory (minimally coupled with matter) with gauge group $G$. My question arises because I am studying the so called Quantum Double models introduced by Kitaev. I understand the algebra of the operators that make the model is isomorphic to that of $D(G)$, what is not clear is the reason why the excitations are classified by the irreducible representations and the conjugacy classes of the quantum double $D(G)$. Any help would be greatly appreciated, also if anyone can give references where I can find some insights on the relation between the algebraic structure of $D(G)$ and Gauge Theories.
Strictly speaking, the statement in your question is not quite accurate: the excitations in the quantum double are classified by irreducible representations of D(G), which is a Hopf algebra. Then one can further say that these irreducible representations can be described using conjugacy classes of G and irreducible representations of the centralizer, etc.
Now, to gain some intuition about why the irreducible reps. of D(G) matter here, it is useful to think about D(G) as the "emergent" symmetry for the low-energy physics, in the sense that D(G) is really the operator algebra of local operators. Particle types by definitions are not changed by application of local operators, so we need to study the representations of the operator algebra so that we can identify the "superselection" sectors. This is obviously far from a complete and precise description of what is actually going on, but hopefully gives some sense why the algebraic structure matters. The best reference on this, as far as I know, is still Kitaev's original paper on quantum doubles.
I'm not sure what kind of relation you are expecting between D(G) and discrete gauge theory, but the following perhaps helps: the quantum double model can be considered as a microscopic realization of a discrete gauge theory. This is most easily seen for the Z2 case, where one has spins living on the edges of a square lattice. It starts off as just a spin model, but now we can look at the Hamiltonian. We can first assume that the "star", or the vertex term, is much stronger than the other one. If one stares at the vertex term it is really nothing but the Gauss's law for discrete gauge fields (in the absence of gauge charges), or in other words if we impose this term locally, we have effectively "local" symmetries of the system. This way we can identify the "low-energy" Hilbert space as that of a Z2 gauge theory.