Compactness has several equivalent definitions, and each one seems designed for a different proof. This note follows the open-cover definition and asks what finite subcovers are really buying us.
A compact space refuses to require infinitely much independent information at once. An open cover can be enormous, but the fact that it covers the space always has a finite witness.
That is why continuous images of compact spaces behave so well: the finiteness can be transported through the map.