In the context of linear representations of finite groups, we know that $$\dim V^G=\frac{1}{|G|}\sum_{g\in G}\text{tr } \rho(g)= \frac{1}{|G|}\sum_{g\in G}|V^g|,\qquad\qquad\qquad (1)$$ where $V^G$ is the vector subspace of $V$ given by $$V^G:=\{v\in V\:|\:g\cdot v=v\text{ for all } g\in G\}$$ and $V^g$ is the set of fixed points by $g$.
By Burnside's lemma, it is clear that $\dim V^G$ is the number of orbits in our representation.
Could someone clarify this a bit? I don't see this as obvious. Is there any other way of seeing that besides using Burnside's lemma and equation $(1)$?