I liked a few results. . For example, smooth analysis by Spielman showed why so many algorithms works very well in practice even though their worst case complexity is terrible e.g. simplex method. Second, concave-convex optimization shows that you will definitely reach closer to global optimum at each iteration. I guess its comforting to know a reason algo behave the way they do.

Sampling theorem is also case in point. Also compressed sensing. Information theory will tell you there is no point trying any harder at compression when you reached epsilon closer to the theoretical limit.