On the Euler number of an orbit space

I like to think of it as a (kind of) categorification of Burnside's formula. If $X$ is a finite $G$-set, then Burnside's formula says that $$ |X/G|=\frac{1}{G} \sum_{g\in G} |X^g|. $$ If $X$ is (homeomorphic to the geometric realization of) a finite $G$-simplicial set, then you can apply Burnside's formula degree-wise, then take alternating sum to obtain your formula for Euler's number. Here you use that both strict orbits and fixed points commute with geometric realization.

For a general $G$-manifold you can realize it as the geometric realization by taking an equivariant triangulation (those exist at least for smooth $G$-manifolds).

Here is a variant of Greg Arone's fine answer.

Following tom Dieck [p.227 of his book Transformation Groups], call a function $\chi$ from finite $G$-CW complexes to $\mathbb Q$ (or any abelian group) additive, if $\chi(X) = \chi(Y)$ if $X$ and $Y$ are $G$-homotopy equivalent, $\chi(\emptyset) = 0$, and $\chi(X \cup Y) = \chi(X) + \chi(Y) - \chi(X \cap Y)$. Because of how one builds $G$-CW complexes, an immediate lemma is that two additive functions agree on all $X$ if they agree on all the single orbit $G$-sets $G/H$.

Both the left and right side of your putative equation are easily seen to be additive functions. Thus they are equal if they are equal on finite $G$-sets, and this is Burnside's formula.