How can a red giant become brighter?

A glance at an H-R diagram will show that red giant stars are post main sequence stars, meaning that they have begun fusing heavier elements than hydrogen in their core.

A star is in constant equilibrium between gravity pulling it toward its center, and heat from fusion blowing it out into space. When hydrogen fusion slows down, gravity begins to collapse the core of the star inwards, causing it to heat up until helium fusion begins. Helium fusion continues to take place at this increased temperature.

Meanwhile, the rest of the star, heated by the core, begins to expand outward, and the surface of the star cools. A cooler gas is less bright than a hotter gas.

However, the star has a much more surface area then before, resulting in an overall increase in the luminosity of the star.

#L = 4 pi R^2 sigma T^4#

In terms of the luminosity equation above, the temperature, #T#, goes down a little, but the radius, #R#, increases a lot (for example our Sun is expected to expand out to the orbit of Mars), so the overall luminosity, #L#, increases. Therefore, even thought the surface of the star is cooler, it will appear brighter.