Compactness is crucial to many discretization arguments. For example, if you have a compact subset K of a domain D in the complex numbers, it is sometimes useful to cover it with a grid of squares. To do this, you argue by compactness that there is some positive delta such that every point in K is at least delta away from the complement of D. Then you place down a grid of squares of sidelength delta/2 (say), and each square that intersects K will lie entirely in D. One reason this is useful is that the boundary of the union of squares that intersect K is guaranteed to be reasonably nice in a way that the boundary of K is not.