I assume that just means the wave interpretation of light. Well if you imagine an infinite number of point sources fill the slit, each one emitting radial waves. These interfere in such a way that a diffraction pattern is produced. Because the points are all located at different points in space there will be phase differences between the waves emitted from each point and hence diffraction occurs.
The diffracted wave contains a different k spectrum because it now contains spatial information about the object that caused it to diffract. Smaller objects (or more correctly, their aperture function along a specific direction) contain more spatial frequency components in that direction, and thus result in a diffracted wave possessing a wider k spectrum.
Since we usually speak of a monochromatic incident wave, the magnitude of k remains constant, thus a change in the components of k imply a change in direction of the wave.
Diffraction is a phenomenon which envolves the bending of waves around obstacles. It's generally guided by Huygen's Principle, which states: every point on a wave front acts as a source of tiny wavelets that move forward with the same speed as the wave; the wave front at a later instant is the surface that is tangent to the wavelets.