What is the reason for phase separation?
The change in Gibbs energy has to be negative for a process to happen (at constant pressure in the absence of work). Gibbs energy is a combination of entropy and enthalpy.


  I do not understand why this phenomenon occurs as a counter example to the natural laws of diffusion along the concentration gradient.


Diffusion is driven by entropy changes, specifically the entropy of mixing (in an ideal solution, the enthalpy of mixing is zero, so the entropy term is the only consideration). The entropy goes up when a solute moves from high to low concentration.

Phase separation occurs when the enthalpy term dominates. If substances A and B are initially mixed and then undergo phase separation, some change occurred (change of temperature, added third component, etc.) so that now phase separation more exothermic (lower energy for [A to interact with A and B interact with B] than [A interacting with B]). Under these circumstances, phase separation can happen.