To address the stated subtext of your post: many people, myself included, take the position that groups are important because they act on things. A representation is just a group action on a vector space (by linear operators). And whenever you have a group action, even if it isn't on a vector space, there is often a closely related representation lurking nearby.
For example, if $G$ is acting on a finite set $S$ then there is an induced representation on $\mathbb{F}^S$ for any field $\mathbb{F}$. More generally, if $G$ is acting on a space $X$ then there is an induced representation of $G$ on the algebra $\mathcal{O}(X)$ of functions on $X$, whatever we mean by "function" in this situation (e.g. perhaps $X$ is a topological space on which $G$ acts by homeomorphisms, and $\mathcal{O}(X)$ is the algebra of continuous real-valued functions on $X$).
To elaborate on this last point: suppose $X$ is a set on which $G$ acts. For any set $Y$, there is an induced action on the set $Y^X$ of maps from $X$ to $Y$: for $g\in G$ and $f:X\to Y$, the map $g\cdot f$ is defined by
$$(g\cdot f)(x) = f(g^{-1}x).$$
If we equip $X$ with some additional structure, like a topology, we may insist that the action of $G$ preserve this structure. This guarantees that the action of $G$ on maps will take structure-preserving maps to structure-preserving maps. As an example, if $X$ and $Y$ are topological spaces and $G$ acts on $X$ by homeomorphisms, then the action of $G$ on $Y^X$ sends continuous maps to continuous maps. In particular, taking $Y=\mathbb{R}$, we have an action of $G$ on the algebra of real-valued continuous functions.
RE: the comments: representations of groups and quivers are both special cases of representations of categories. If $R$ is a ring, then an $R$-rep of a group is a rep of its $R$-group algebra; an $R$-rep of a quiver is a rep of its $R$-path algebra; you can view an $R$-algebra as an $R$-linear category with one object; and the category of modules over an $R$-algebra is an $R$-linear (typically abelian) category.
Everything you see in a first course in representation theory can be encoded in category theory, if you really want to. Maschke's theorem says that (if $R$ is a field of characteristic coprime to the order of $G$) the category $C$ of $R$-reps of $G$ is semisimple. There's a pairing on the objects of $C$ defined by $\langle W,V\rangle= dim\ Hom_C(W,V)$, and this is the usual pairing on characters. Frobenius reciprocity is the fact that restriction and induction are adjoint, as mentioned in the comments. The other main results have similar interpretations.
In such a basic setting, this is all very uninteresting, but much of current research in representation theory involves understanding properties of categories of representations. The reason that you never really see people go to the effort of doing semisimple reps of finite groups categorically is that there isn't much reason to. The strength of category theory is really in organising abstract results, and suggesting ways that you might study some new objects that you might not know very much about. I'd suggest that the better way to think about things might be to treat translating the main theorems of basic representation theory of groups into categorical language, and then having this category as an excellent example of an abelian category. This can then give you a good toy example for the general theory!
Edit: I should say that the abstract rep theory of finite groups over "good" fields is "completely categorical", but this is only when you look at "general facts", by which I mean general theorems true for any finite group. But most research focuses on specific (classes of) groups, e.g. finite groups of Lie type. There are lots of phenomena which might not have such a clean categorical interpretation (e.g. reducibility of parabolic induction, Deligne--Lusztig theory); typically these kinds of results imply strictly weaker categorical results (in these examples, they tell you about block decompositions of categories), because they include non-categorical information -- they give explicit representatives of isomorphism classes of objects. In other words, once you want to get your hands dirty and do some useful representation theory, you'll probably have to leave the world of category theory!
Also, a couple of words of warning: once you leave the world of finite groups, or once you allow your coefficient ring $R$ to become sufficiently "bad", things can get more difficult; you start to encounter topological issues, which might make the category theory a bit more unpleasant.
Best Answer
I'll consider complex representations.
When I learned representation theory for the first time, it took me a while to realize why that definition makes sense, so I'll try to give an intuitive explanation.
I guess the sentence is from the definition of the regular representation of a group in section 1.2 (b) in Serre. Abstractly the vector space defined there is isomorphic to $\mathbb{C}^n$ and (in principle) $G$ just serves as the indexing set. Take some bijection from $G$ to $\{1, \ldots, n\}$ and you have the usual basis indexing. However, there is a good reason to index the basis by elements of the group:
As you have seen, the idea of a representation is to consider the group $G$ as acting on a finite dimensional vector space i.e. to consider the elements as matrices. Quite early on, one might ask if every group has such a representation and what restrictions there are and if there is a natural choice for a vector space and action etc. Indeed, there are two (somewhat) natural choices.
The first choice is the trivial representation (defined just in 1.2 (a) in Serre's book) where one represents every $g \in G$ by the $0$-matrix in $\mathbb{C}^{1 \times 1}$. The second one is more interesting:
Every group acts on itself by left multiplication $G \times G \rightarrow G: a.g \mapsto ag$. Now consider $\mathbb{C}^n =: V$ and associate to each element $g \in G$ a basis vector $e_g$ and define for any element $a \in G: a.e_g = e_{ag}$. This gives an action on $\mathbb{C}^n$ that is inherited by the groups action on itself. The elements $V$ are of the form $\sum_{g \in G} \lambda_g e_g$ and the explicit matrix representing an element $g \in G$ is entirely given by how $g$ acts on the other elements in $G$.
Notice now that we have a "natural" way of interpreting $G$ as a group of matrices. However, matrices can sensibly made into an algebra over $\mathbb{C}$ i.e. it does not only make sense to consider the action of $a \in G$ on an element $\sum_{g \in G} \lambda_g e_g$ but also to consider the action of $\sum_{a \in G} \lambda_a e_a$ on an element $\sum_{g \in G} \lambda_g e_g$. This action is uniquely determined by the action og $G$ on itself and the fact that that action on an element of $V$ needs to be linear. Ultimately this gives
$$ (\sum_{g \in G} \lambda_g e_g) \cdot (\sum_{h \in G} \mu_h e_h) = \sum_{g \in G} (\sum_{h \in G} \lambda_{gh^{-1}} \mu_h) e_g = \sum_{g,h \in G} (\lambda_g \mu_h) (e_{gh})$$
Compare this now to discrete convolution and notice why indexing by the group elements is sensible. Also explicitely look at the example where $G:=\mathbb{Z}/p \mathbb{Z}$!
Considering the vector space $V$ together with the above defined multiplication makes it into a $\mathbb{C}$ algebra - the group algebra over $G$.
Also, if you are just starting out in representation theory, Serre's book may be a bit heavy/dry (at least it was for my taste). Maybe have a look at James & Liebeck. They explicitely compute many examples and give a more concrete introduction. If you are looking for something more geared towards Lie Theory, consider Fulton & Harris (PDF available online).