Common tangent construction
Several answers (full disclosure - mine) are available that shed at least some light. So read them first.

https://physics.stackexchange.com/questions/196686/what-causes-a-eutectic-point

Plot Binary Eutectic from Temperature of Melting and Enthalpy

OK. Now to your confusion (I think). What does a straight line across a plot of free energy vs composition actually mean? Well, mathematically that straight line represents the weighted average of the pure free energies, $x_{A}G_{A} + x_{B}G_{B}$ (perhaps more obvious using $x_{B} = 1-x_{A}$). So, $G_{A}$ is the intercept at $x_{A}=1$ and $G_{B}$ is the intercept at $x_{B} = 1$. If the free energies are a fixed value, the chemical potentials are a fixed value. So, the tie-line that is a common tangent to two free energy curves is the solution where the free energies (chemical potentials) are equal in both phases. 

As for why the one-phase region does not chemically separate, the question is why should it? There is no second composition that has a common tangent, so there is not way to make the chemical potentials the same in two phases while in a one-phase region.