What happens to make a red giant star redder than a main sequence star?

When most of the fuel is used, the star expands as the pull of gravity inwards is lessened, resulting in a drop in temperature. Redder means less temperature.

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.

A red giant star appears redder than a main sequence star because of its higher surface temperature, which causes it to emit more red light.