Why do post main sequence stars enter the red giants branch?
If your question is why the star starts moving up the red giant branch, it's in essence because of the behaviour of the surface opacity and the development of a substantial convective envelope in order to meet the surface boundary condition. It's basically the Hayashi track in reverse. 

We can say this because if you create a model of a star and artificially suppress convection in the equations, then the model just keeps getting colder and colder at roughly constant luminosity, instead of ascending the giant branch.  See e.g. Fig. 2 of Stancliffe et al. (2009). Note that the star still becomes large in terms of radius, just not bright.