Proof that the inverse image of a single element is a discrete space

Fix $y\in Y$ and $x\in f^{-1}(y)$. Since $f$ is a local homeomorphism, there is a neighborhood $U$ of $x$ such that $f|U : U\to f(U)$ is a homeomorphism. If $z\in U\cap f^{-1}(y)$, then $f(z) = y = f(x)$; since both $z, x\in U$, injectivity of $f|U$ implies $z = x$. Therefore $U\cap f^{-1}(y) = \{x\}$. As $x$ was arbitrary, $f^{-1}(y)$ is discrete.

Let $f:X \to Y$ be a local homeomorphism.

Suppose $y \in Y$ and let $x \in F_y:=f^{-1}[\{y\}]$, the fibre of $y$.

Then $x$ has an open neighbourhood $U_x$ such that $f|_{U_x}: U_x \to f[U_x]$ is a homeomorphism (by the definition of being a local homeomorphism).

In particular, $U_x \cap F_y = \{x\}$ or else we have some $x' \in U_x \cap F_y$ which means $f(x')=f(x)=y$ while $x,x' \in U_x$ contradicting the fact that $f|_{U_x}$ is injective (being a homeomorphism). So $U_x$ witnesses that $x$ is an isolated point of $F_y$, showing that $F_y$ is indeed discrete in the subspace topology.

Note that local injectivity is all we need.