What happens to make a red giant star redder than a main sequence star?
Red giants have huge size.
So heat is radiated by large surface area and so the temperature drops..When most of fuel is finished star expands as pull of gravity inwards is reduced,.less temperature means red color.
I red giant is redder because it is larger and the outer layers are cooler.
When a star of less than about 8 solar masses runs out of Hydrogen in its core, the core starts to collapse under gravity. The now mainly Helium core heats up to the point where Hydrogen fusion starts in the layers around the core.
This Hydrogen fusion causes the star to expand massively. To put it in perspective, when our Sun becomes a red giant it will expand to around the size of the Earth's orbit and will consume Mercury, Venus and probably Earth.
As the star becomes so big that its outer layers cool down to less than 5,000K. This makes the emitted light from the surface redder.