Why do red giant stars have low temperatures?
As Tham has explained, red giants expand because their cores are much hotter than dwarf stars. This expansion gives them a much larger area to radiate their energy from. A star’s photosphere (the surface that we can see) acts as a black-body radiator, which means that the energy radiated per unit area is proportional to the fourth power of the surface temperature, and the peak wavelength decreases (becomes bluer) with increasing temperature. The net result is that even though the star is putting out more energy than before, it is spread over so much larger a surface area that the temperature at which it radiates is lower, and the peak wavelength drops into the red.
