Comment by MajimasEyepatch

As an example, if you're trying to "smooth out" some signal, what you're really trying to do is remove the high frequency components. So using a Fourier transform to convert it to the frequency domain lets you directly work with the frequency data, rather than indirectly in the time/space/whatever domain. The fact that the operation is simpler in the frequency domain is a good hint that you've picked the "natural" space in which to look at the problem. Of course, there's no formal definition of "naturalness," and at the end of the day, you get the same result either way.