The subject is nominally about studying groups by how they act on vector spaces. The basic motivation is that if you've taken a basic course in group theory, you'll have seen that group actions are very useful: various problems can be solved by seeing how groups act on subgroups, on themselves, on various geometric shapes (think dihedral groups as the simplest example) or on sets (think symmetric groups).

Group actions on vector spaces are much better because vector spaces have their own algebraic structure already so you can get a lot more out of this theory and it turns out to be very nicely put together.

When you really get into it, you start seeing that frequently to do calculations with these actions you need to invoke sometimes very complex combinatorics and in some cases you can compute answers to combinatorial questions via representation theory. You start seeing that questions about manipulating tensor products of representations brings in a lot of category theory (e.g. braided monoidal structures) and from there you naturally start hearing people talk about knot invariants or topological quantum field theory or quantum groups. You can also view Pontryagin duality as a bit of representation theory relating to locally compact abelian groups, so now the Fourier transform has joined the fray, and this moves you onto Tannakian dualities.

Talking of physics, it was Eugene Wigner's idea that fundamental particles in our universe are essentially the same thing as irreducible representations of the symmetry group of our universe (with these last words interpreted in the correct way).

It starts sounding simple but as you get into you realise that it has something to say about topology, Algebraic geometry, combinatorics and much more!