For me personally, the first time I "got" Fourier analysis was when I understood how it could be used to prove Roth's theorem on arithmetic progressions (that any dense set of integers contains an arithmetic progression of length 3). It can be proved in several ways, but when you see this way you immediately realize that the technique is likely to be useful for many other problems. And in fact, Roth's theorem can also be used to justify Szemerédi's regularity lemma (which is not quite the same as justifying a whole theory, but it is a very useful technique) in a similar way.