This follows from the following regularity estimate for the flat Laplacian case (which is, I believe, proved in Warner's book using Fourier series on a torus but also in most standard texts on elliptic PDEs): Given a bounded open domain $\Omega \subset \mathbb{R}^n$, there exists $C>0$ such that for any function (or even just a distribution) $u$ compactly supported in $\Omega$,
$$ \tag{*} |u|_{H^2} \le C|\Delta_0 u|_{L^2}, $$
where $\Delta_0$ is the standard flat Laplacian.

To extend this to a local regularity estimate for the Laplace-Beltrami operator, it suffices to prove regularity estimate for $u$ compactly supported on a neighborhood of each point $p \in M$. If you use geodesic normal coordinates on a sufficiently small neighborhood of $p$, then you can assume that the Laplace-Beltrami operator is of the form
$$
\Delta u = (\delta^{ij} + a^{ij}(x))\partial^2 + b^k\partial_ku
$$
where $|a^{ij}|, |b_k| < \epsilon << 1$.
Therefore, if $\Delta_g u = f$, then
$$
\Delta_0u = -a_{ij}\partial^2_{ij}u - b^k\partial_ku + f
$$
Therefore, by $(*)$
$$
|u|_{H^2} \le C(\epsilon |u|_{H^2} + |f|_{L^2}).
$$
If the neighborhood is sufficiently small, then $C\epsilon < 1$ and therefore,
$$
|u|_{H^2} \le C|f|_{L^2}.
$$

Partial Differential Equations IandPDE IIby Taylor. He develops the theory on manifolds. – Neal Dec 7 at 2:26