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The reason is due to the different production mechanism for the energy output from the star and the increased size of the Red Giant.

As a star moves into its Red Giant phase, after not being able to maintain gravitational equilibrium due to hydrogen begin exhausted in the core, the star begins to collapse. This collapse heats a hydrogen shell just outside the core (previously not available for fusion) initiating fusion of hydrogen to helium in the shell. This means fusion is taking place closer to the surface of the star and leads to increased reaction rates then before. This produces enough energy to increase the star's luminosity by a factor of 1,000--10,000. The temperature of the star goes down because the star balloons out with this increased reaction rate, meaning the energy is distributed over a larger area. Hence, you get a bright, cooler star.

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