Picture below is from 293 page of Huisken, Gerhard, Asymptotic behavior for singularities of the mean curvature flow, J. Differ. Geom. 31, No.1, 285-299 (1990). ZBL0694.53005.
$A$ is second fundamental form. $H$ is mean curvature. Why at a point we have $|A|^2=H^2$, then we have it on $M$ ?
On the previous page, Huisken shows that for a compact hypersurface satisfying $H=\langle \mathbf{x},\mathbf{\nu}\rangle \ge 0$ we have $|A|^2 = \alpha H^2$ for some constant $\alpha$. Thus if $|A|^2 = H^2$ at one point, we know $\alpha = 1$ and so this equation holds everywhere.
This relies heavily on the hypothesis of the self-similarity condition - you can't deduce it from the information you've provided in your question.