Many formal definitions of "representation of a group" give no indication at all of why we might care, apart from the possibility of defining things and proving lemmas and theorems about them.
To my perception, there is a genuine utilitarian aspect to "representation theory", apart from self-referential questions. Namely, groups act on physical spaces, and, thereby, act on functions on those spaces. (This is an idea going back many decades, espoused by Gelfand and others...) If we know all the irreducible repns of the given group, we may hope to express every function on such a physical space as an infinite-sum-or-integral of elements of irreducible representations. In particular, if/when we have extra understanding of the elements of the irreducibles as manifest in that space of functions, we can find out many further things.
The innocuous examples of this are Fourier series, Fourier transforms... which are already very serious examples of representation theory of $\mathbb R^n$, and also spherical harmonics... (the action of orthogonal/rotation groups, as well as dilations).
(I think it is rarely about somehow clarifying the structure of a group itself by talking about its representation theory. In the cases I care about, the apparent physical structure of the group is clear, at an immediate level. The subtlety of its representation theory is often quite surprising! E.g., $SL(2,\mathbb R)$.)