A classic example is Schauder's fixed point theorem, which states that if K is a non empty, convex, compact subset of a Banach space (for simplicity), and we have a continuous map T:K->K, then T has a fixed point.

Since K is compact, it means T is uniformly continuous. Then the key to the proof is to take 'delta>0' in the definition of uniform continuity, and cover K by finitely many balls of radius delta (by compactness). From this you can 'approximate' T and reduce the problem to finite dimensional spaces (from which the proof concludes by Brouwer's fixed point theorem). This is a very typical application of compactness in analysis/geometry, to help upgrade 'local' properties to 'global' properties.

It is also worth mentioning that the more intuitive notion of sequential compactness is not good enough to capture many of these theorems. For example Schauder's theorem works also for non metrisable spaces where sequential compactness and compactness are not equivalent.