Anisotropic Diffusion Filter - Intuition Behind Parameters
This paper - The Structure of Images describes quite nicely to get to the diffusion equations from the point of view of a filter. It's about the isotropic case, but anisotropic is really just an extension of that. For anisotropic diffusion see Scale Space and Edge Detection Using Anisotropic Diffusion as Deniz suggests this one is quite good as it also includes discretised versions of the equations.

The basic idea is to solve the heat equation where $t$ is the scale of the image, i.e. the amount of filtering to apply. For anisotropic diffusion the diffusion coefficient $c$ is replaced by $g$ which is dependent on the image gradient and therefore filters preferentially depending on the gradient. 

The divergence here really just arises from the definition of the heat equation.