Consider an isometry from a compact metric space to itself, this is simply a function that preserves distances. Notice that this implies that the function is continuous and moreover that it is injective. What we have to prove is that such a function is also surjective. Observe that this implies that the function is indeed a homeomorphism.
A consequence of this fact is that the category that has compact metric spaces as objects and isometries as arrows enjoys a Cantor-Schröder-Bernstein-like property.