To prove this, use naturality of the Leray-Serre spectral sequence: the fixed point has a $G$-equivariant map $\ast \to M$ that induces a map of spectral sequences $$ H^p(G;H^q(M;\Bbb Z)) \to H^p(G;H^q(\ast;\Bbb Z)). $$ (This converges to the map $H^*_G(M) \to H^*_G(\ast)$.)
This map is compatible with the differentials, the map is an isomorphism when $p=0$, and the target is zero when $p > 0$. If an element $y$ on the 0-line were in the image of a differential, $y = d_r(x)$, then applying this map of spectral sequences we'd find $y = d_r(0)$ and hence $y=0$.