The following answers might be useful:

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