To find the charge distribution of proton , we study electron proton scattering and compute the form factor to find the cross section. The form factor comes out to be Fourier transform of charge distribution. It is then multiplied by the cross section calculated under assumption that proton is point particle. What I don't understand is why the form factor has to be Fourier transform of charge distribution. I don't want the derivation but some intuition.
You have surely seen slit experiments, where waves which pass through the slit and scatter to produce a pattern on a screen placed behind the slits. In the far-field approximation (also known as Fraunhofer diffraction) this pattern is precisely the Fourier transform of the slits. Perhaps you even remember that waves passing by lines produce the same pattern as waves passing through slits, given that the line and the slit have the same dimensions.
We can also play this game with holes and points or any kind of shape, really. The proton charge distribution is such a shape and the electron is a quantum mechanical wave passing along it. Like every other wave passing along an object, the wave will scatter and it is described by the Fourier transform of the object. The far field approximation always holds in a realistic experiment, because the the charge distribution is far smaller than the travel distance of the electrons.