Classification of $\operatorname{Rep} D(G)$
There are some classic results on the classification of the irreducible $D(G)$-modules:
If the field is the complex numbers $\mathbb{C}$, it has been shown that a representation of the finite group $G$, induced from an irreducible representation of the centralizer subgroup of an element $g$ of $G$, generates an irreducible rep of $D(G)$ and that furthermore, all the irreducible quantum double modules are obtained in this way. Proofs for these results can be found at:
Quantum double finite group algebras and their representations, Bull. Austr. Math. Soc., 48, 1993, p.275-301, by M.D. Gould.
(See section 6, mainly theorem 6.3).
In there, it is also shown that all such algebras are semisimple and their character theory is developed.
From a more general viewpoint, representations of $D(G)$ over algebraically closed fields of arbitrary characteristic have been studied at:
The representation ring of the quantum double of a finite group, J. of Algebra, 179, p.305-329, (1996), by S.J. Witherspoon. In there, some of the previously mentioned results have been generalized: for example an analogue of Maschke's theorem is proved; it is shown that $D(G)$ is semisimple if and only if the characteristic $p$ of the field, does not divide the order of the group $G$.
Furthermore, the representation ring $R\big(D(G)\big)$ of the quantum double is studied: it is shown to be a commutative algebra, a direct sum decomposition is described and a classification of the indecomposable $D(G)$-modules is also achieved (among other results as well).
There's a higher way to come at this. I'll be a little light on the rigorous details here, but everything I mention can be found in the book "Tensor Categories" by Etingof, Gelaki, Nikshych, and Ostrik. The book's a very good starting point for moving from the Hopf algebras perspective to the tensor category perspective, which is where a lot of current research is being done.
If one considers a semisimple Hopf algebra $H$, and take $\mathcal{C}=\operatorname{Rep}(H)$ to be the category of finite dimensional left (or right) modules of $H$, then there is a braided tensor equivalence $\operatorname{Rep}(D(H))\cong \mathcal{Z}(\operatorname{Rep}(H))$, where $\mathcal{Z}(\mathcal{C})$ denotes the categorical center of the category $\mathcal{C}$. This center construction works not just for the particular choice here, but any tensor (aka monoidal) category with sufficiently similar properties. The objects of the center are the pairs $(V,\gamma_V)$ where $V$ is an object of $\mathcal{C}$ and $\gamma_V$ is a natural family of isomorphisms called a "half-braiding" (because they piece together into a braiding on the entire category).
In the case of $H=\mathbb{C}G$ with $G$ a finite group, we can go one step better. There is a Morita-equivalence between $\operatorname{Rep}(G)$ and $\text{Vec}_G$, where the latter is the space of $G$-graded finite-dimensional vector spaces (over $\mathbb{C}$). This is equivalent to saying these categories have the same centers, up to braided tensor equivalence, so we could just as well compute $\mathcal{Z}(\text{Vec}_G)$ instead. Once you actually write down what the half-braiding conditions are, this center becomes very easy to determine: it's $\text{Vec}_G^G$ (sometimes denoted ${}^G_G\mathcal{M}$, or some variation thereof depending on the use of left/right (co)actions), the category of finite dimensional $G$-graded, $G$-equivariant vector spaces. At this point it's easy to decide the isomorphism classes of the irreducibles, and you find that they are parameterized by pairs $(g,\chi)$ where $g$ is an element in a complete set of representatives of the conjugacy classes of $G$, and $\chi$ is an element in a complete set of representatives for the irreducible representations (or characters) of $C_G(g)$. So the isomorphism type of the module depends only on the conjugacy class of $g$ and the isomorphism class of $\chi$.
When you understand the objects of $\text{Vec}_G^G$ it becomes readily apparent that the irreducible objects are just induced representations from $C_G(g)$ to $G$, but where the implicit grading of this induction via cosets of $C_G(g)$ is relevant to deciding the full action of $D(G)$.
And if you want to go even further than that, you can change the associativity morphism of $\text{Vec}_G$ via a normalized 3-cocycle $\omega$ to obtain the category $\text{Vec}_G^\omega$, and then we have $\mathcal{Z}(\text{Vec}_G^\omega)\cong\operatorname{Rep}(D^\omega(G))$, where $D^\omega(G)$ is the twisted Drinfeld double, and is in general a quasi-Hopf algebra and not a Hopf algebra. These objects are also quickly described in the paper you mention. The description of the irreducibles is similar, except now we're using irreducible projective representations for particular 2-cocycles of $C_G(g)$ obtained from $\omega$.
This category, as a braided tensor category, will only depend on the cohomology class of $\omega$, while $D^\omega(G)$ can have wildly different structures even for representatives of the same cohomology class. Since those structures are also quite nightmarish to deal with directly for any non-trivial 3-cocycle, most people end up gravitating towards dealing with them through their representation categories, instead.
This is a study note that spells out @Konstantinos's answer explicitly.
Preface
Our goal is to classify all finite dimensional representations over the complex number field for the quantum double $D(G)$ for a fixed finite group $G$, with proofs. We will use [G] as out main reference, while auxiliary results can be found in [S] and [CR].
For other considerations, see [W], [L], and [B]. For the representation theory of $D(G)$ over other fields, see [W]. For the representation theory of other Hopf algebras, see [L], which deals with a class of (possibly infinite dimensional) Hopf algebra with a technical condition: co-semi-simple + involutive). For more applications, see [B].
Abstract
In what follows, semi-simplicity allows us to focus on the simple modules. We can get lots of them by induction from the underlying group $G$. Character theory for $D(G)$ distinguishes the simple modules we get from induction, showing the abundance. The structure theorem of $D(G)$ predicts how many non-isomorphic simple modules there should be. Examining how many different simple modules we've got from allows us to complete the proof.
Semi-simplicity of $D(G)$
Following [G] and its notations, the first main result is the semi-simplicity of $D(G)$. Theorem 2.3 says that any finite dimensional Hopf algebra $A$ is semi-simple if and only if there exists a left integral $x \in A$, this is a powerful criterion for semi-simplicity. A left integral of $D(G)$ is given in [G. (16)], where $x = E_\iota 1^*$, so $D(G)$ is semi-simple.
The proof of theorem 2.3, the powerful semi-simplicity criterion, can be found in [S. Theorem 5.18]. There, Sweedler first defined the left integrals for $H^*$. As $H$ is finite-dimensional, $H$ is isomorphic to $H^{**}$ naturally, whose left integrals can be doubly-dualed back to $H$. This definition coincides with that of [G]. Anyway, one can use a left integral to "average" an arbitrary linear projection and get a Hopf linear projection from any larger module to any smaller submodule, proving semi-simplicity. An explicit averaging formula is given in the proof of [S. Theorem 5.18]. The other side is easy: if $H$ is semi-simple, than the complement of $ker(\epsilon)$ is the set of left integrals. A few immediate corollaries are
$D(G)$, $\operatorname{Fun}(G)$, and $\mathbb{C}[G]$ are all semisimple.
$k[G]$ is semisimple if and only if $\epsilon(x=\Sigma g) = |G|$ is not zero, which in turn is equivalent to that $|G|$ is not divisible by $\operatorname(char)k.
$k[X]/<X^p>$ is not semi-simple, since $\epsilon(x^{p-1})$ is zero.
$k[X]/<X^p - X>$ is semi-simple, since $\epsilon(x^{p-1})$ is -1.
Unitarity of representations and orthogonality of matrix elements
Every finite dimensional $D(G)$-module is equivalent to a unitary one [G. Lemma 4.1], so in particular $D(G)$ is proven again to be semi-simple. Routine arguments show the orthogonality of matrix elements [G. Theorem 4.1]. Applying this to characters, we get the orthogonal relations among them [G. Theorem 5.1]. Note that this can be generalized to a larger class of Hopf algebras (possibly infinite dimensional), which are co-semi-simple and involutive [L]. The rest of chapter 5 in [G] exhibits the character theory for $D(G)$ and finds an explicit basis for the center of $D(G)$ [G. (25) -- Thm 5.2]. This basis is in 1-1 correspondence to the number of $G$-equivalence classes of $Q$, and is also in 1-1 correspondence to the set of non-isomorphic irreducible $D(G)$-modules by the structure theorem for Artinian semisimple rings [G. Theorem 5.2]. We will justify the last statement later.
Enumeration of representations of $D(G)$
The representations of $D(G)$ can be obtained by induction from the centralizer subgroups of $G$. This is done in chapter 6. The character theory developed in chapter 5 distinguishes one from another, showing the abundance of the results. Since we have known how large $\operatorname{Irrep}(D(G))$ is, we will be done by showing the structure theorem for $D(G)$.
Structure theorem for $D(G)$
In this section, our reference is [CR. section 23 to 26]. From now on, we will assume $R$ to be a unital Artinian ring (associative, but not necessarily commutative). We will show the structure theorem for $R$ if it is semisimple. Since $D(G)$ obviously satisfies all the conditions, we will then be done.
Since $R$ is Artinian, any left ideal $I$ is nilpotent if and only if it has no idempotent elements. It is then easy to show that the set of nilpotent left ideals is closed under finite sum. More interestingly, the sum of all nilpotent left ideals is a nilpotent two-sided ideal, called the radical $\sqrt(R)$ of $R$. If the radical is zero, we call $R$ semisimple. It is easy to show that $R/\sqrt(R)$ is semisimple.
If $R$ is semisimple, then any minimal left ideal $L$ is not nilpotent and thus have an idempotent element $e$. Minimality guarantees that $L$ is generated by that idempotent element. Note that the generator is not unique in general. In this case, $R = Re \oplus R(1-e) = L \oplus L'$. One can furthur decompose $R$ into $R = Re_1 \oplus \cdots Re_n$, where the $e_i$'s are orthonormal idempotents. It is easy to show the uniqueness of decomposition, and also that any $R$ with this decomposition is in fact semisimple. The decomposition breaks the unit $1$ into the sum of the $e_i$'s, this is the key. Using this key, it is not hard to show that every left $R$ ideals are completely reducible [CR. 25.8], and also that any irreducible $R$-module is isomorphic to some minimal left ideal in $R$.
Therefore, the complete set of non-isomorphic simple modules can be found in the decomposition of $_RR$ as a left $R$-module! The Wedderburn structure theorem shows that the number of them is the same as the size of the center of $R$ (TODO: needs clarification). This completes the argument.
References
[G]. Quantum double finite group algebras and their representations, Bull. Austr. Math. Soc., 48, 1993, p.275-301, by M.D. Gould.
[S]. Hopf algebras (Benjamin, New York, 1969), by M.E. Sweedler.
[CR]. Representation theory of finite groups and associative algebras, by C.W. Curtis and I. Reiner.
[W] The representation ring of the quantum double of a finite group, J. of Algebra, 179, p.305-329, (1996), by S.J. Witherspoon.
[L] Characters of Hopf algebras, J. Algebra 17 (1971), 352-368, by R.G. Larson.
[B] Exactly solved models in statistical mechanics (Academic press, 1982), by R.J. Baxter.