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As we look at the center we look directly down into the Sun and see bright hot layers deep in the photosphere. When we look at a limb we see through the top layers at an angle and so do not see as deeply. Thus, we see photons from higher, cooler layers and the limb appears less bright.
Limb darkening allows us to deduce that the temperature decreases with altitude in the photosphere, and to find the rate of decrease if the darkening is measured carefully as a function of radius.