Why Do Red Giant Stars Get Bigger?

During the main sequence phase of solar mass stars they are in hydrostatic equilibrium. This is where the gravitational forces trying to collapse the star are balanced by outward radiative pressure created by fusion in the core. When the core is depleted of hydrogen the now helium core contracts and allows a shell of hydrogen fusion to occur around it. This shell of hydrogen fusion has a large volume and creates a larger outward radiative pressure. The gravitational forces do not change so this is able to cause an expansion of the outer layers, which in turn moves the star into the Red Giant phase of its evolution.