This question is a bit vague mostly because of my very limited knowledge of group theory. But I would like to know if there is some intuitive example of representation theory that gives a layman an 'a-ha' moment.
Let us just consider the group $SO(2)$ and denote the representations by $\left|m \right>$ for $m=0,\pm 1, \pm 2, ...$. Then for some generator $J$ we have $J\left|m \right>=\left|m \right> m$, and for a rotation by $\phi$ we have $$U^m(\phi)\left|m \right>=\left|m \right> e^{-im\phi}$$ where rotation by $\phi$ get mapped to $U^m(\phi)$, $R(\phi) \rightarrow U^m(\phi)$.
Why have all these representations? Don't I just need $m=1$, the so called faithful representation?
Of course there is the Fourier analysis picture and that I get a localized $\left|\phi\right>$ by summing over all these representations, $$ \left|\phi\right>=\sum_m \left|m\right>e^{-im\phi}$$ which is kind of saying "rotate by $\phi$ in all representations", is that expressing $\phi$ as group member (or an 'action') as opposed to just a plain angle? Obviously something is gained by having these degenerate representations $-$ but shouldn't all the information be already available in the faithful representation?