Why does Fourier analysis of Boolean functions "work"?

Over the years I have gotten used to seeing many TCS theorems proved using discrete Fourier analysis. The Walsh-Fourier (Hadamard) transform is useful in virtually every subfield of TCS, including property testing, pseudorandomness, communication complexity, and quantum computing. While I got comfortable using Fourier analysis of Boolean functions as a very useful tool when I'm tackling a problem, and even though I have a pretty good hunch for which cases using Fourier analysis would probably yield some nice results; I have to admit that I'm not really sure what it is that makes this change of basis so useful. Does anyone has an intuition as to why Fourier analysis is so fruitful in the study of TCS? Why so many seemingly hard problems get solved by writing the Fourier expansion and performing some manipulations? Note: my main intuition thus far, meagre as it may be, is that we have a pretty good understanding of how polynomials behave, and that the Fourier transform is a natural way of looking at a function as a multilinear polynomial. But why specifically this base? what is so unique in the base of parities?