While one key fact, about the trace of projection being the dimension of the subspace, was already mentioned, I think it is also important to mention another key fact about traces that is useful for characters. That is that the trace defines a natural inner product on the space of linear transforms.
Indeed, there is a natural inner product on $\operatorname{Hom}(V, V)$ defined in the coordinate form as
$$ \langle A, B \rangle = \sum\limits_{i, j} A_{ij} B_{ij}^*. $$
But more importantly, in the coordinate-agnostic form, this same inner product is expressed as
$$ \langle A, B \rangle = \sum\limits_{i} \left(\sum\limits_j A_{ij} B_{ij}^*\right) = \sum\limits_{i} (AB^*)_{ii} =\operatorname{tr} [AB^*]. $$
What's nice about it is that we can apply it to the representation $\rho : G \to \operatorname{Aut} V$, and get
$$ \langle \rho(g_i), \rho(g_j) \rangle = \operatorname{tr}\rho(g_i \circ g_j^{-1}) = \chi(g_i \circ g_j^{-1}). $$
Then, using the inner product on the regular representation, one may find that
$$ \frac{1}{|G|} \langle \rho(g_i), \rho(g_j) \rangle = \begin{cases} 1, & i=j, \\ 0, & i \neq j, \end{cases} $$
from which a lot of things derive, such as the Plancherel formula and the inverse Fourier transform.