The fact is not valid for general topological spaces. For example, the identity $i:\mathbb{R} \rightarrow \mathbb{R}$, when $\mathbb{R}$ is with the antidiscrete topology (only the empty set and the whole $\mathbb{R}$ are open). It is a local homeomorphism, but the inverse image of a point is not closed (but is discrete).
The discretude of inverse image of points is valid in general: each point has an open neighborhood that makes the aplication homeomorphism, then bijection, and not two points of $f^{-1}(y)$ can lie on this neighborhood.
When the singleton $\{y\}$ is closed, then $N$ $\backslash$ $\{y\}$ is open. $f^{-1}(N$ $\backslash$ $\{y\}) = M$ $\backslash$ $f^{-1}(\{y\})$ is open and follows that $f^{-1}(\{y\})$ is closed. Note that the hypothesis that $\{y\}$ is closed is not just sufficient, but necessary when $f$ is surjective. Because $f^{-1}(\{y\})$ closed $\Rightarrow M \backslash$ $f^{-1}(\{y\})$ open, and because an local homeomorphism is an open map, $N$ $\backslash$ $\{y\}$ is open, then $\{y\}$ is closed.