The following answers might be useful:

* [Fourier-Mukai transform - a first example](https://mathoverflow.net/questions/5726/fourier-mukai-transform-a-first-example)
* [Intuition for Integral Transforms](https://mathoverflow.net/questions/2809/intuition-for-integral-transforms)
* [Fourier transform for dummies](https://mathoverflow.net/questions/446/fourier-transform-for-dummies)

The last one has my sketch of an answer which I'll post here once it gets better.