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Red giant surface is HUGE. And because it is huge, it requires that (lower) temperature to radiate that amount of heat (energy) produced in the core.
It is just equilibrium.....
If you try to squeeze the star, its surface will become smaller and it will not be able to get rid of this power, temperature will go higher so it has to expand...
The equations for black body radiation deal with these issues...
I would have to dig out those details from my uni notes :-)
but I remember the calculation is fairly straightforward, if you know power output from core for "normal" and degenerate core cases...
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