Yes, this provides good intuition about why it is useful: the PDF of the sum of two random variables is the convolution of the original PDFs. A convolution is awkward to work with, but by the convolution theorem it is a multiplication in the Fourier domain. This immediately suggests that the Fourier transform of a PDF would be a useful thing to work with.
If you don't say that this is what you are doing then it all seems quite mysterious.
If you don't say that this is what you are doing then it all seems quite mysterious.