Why do electrons that are closer to the nucleus have more negative energy?

Question

And how does that make the electron more stable?

Answer 1

The same reason that an object resting on earth has more negative energy than one at a high elevation. In any inverse square law field (like electric and gravity), the only meaningful place to define an energy level of zero (reference) is at a distance of infinity.

An object on a mountain top can fall to earth to achieve a lower energy state. The object already resting at sea level has nowhere to fall. An electron at a higher energy level can drop to a lower one, emitting a photon. An electron at its lowest energy level has nowhere to drop.

Answer 2

Don't think of it in terms of more negative energy. Just think of it in terms of having less overall energy. It requires more energy for the electron to keep a larger distance from the nucleus, since more is required to overcome the attractive forces.

As far as stability, all atoms are more stable when they have all the lower orbitals filled by the electron (think of it as having a solid base/foundation). Thus, if an electron has extra energy and is in a higher orbital with a lower orbital empty, it will ahve a tendency to shed the extra energy (in the form of a photon) and bounce to the lower orbital.