Stability of energy levels in quantum mechanics

Quantum mechanics per se cannot explain this behaviour. The Schrödinger Equation predicts that an electron in any energy level should remain there at any time in the future (as long as no other interactions occur).

Quantum electrodynamics however introduces the concept of spontanous emission: the electron falls into a lower energy level and emits a photon of the corresponding energy difference. At the lowest energy level, this is not possible, since there is no lower state to fall into, so this ground state is always stable.

Depending on the spacing of energy levels, some levels may be more stable than others, but the ground state is always the most stable one.

Even before QM, there was the minimum total potential energy principle. This states that for example, in the simplest case, a free electron and proton will combine into the lowest possible energy state possible, and this is the most stable configuration.

A free proton and free electron will tend to combine to form the lowest energy state (the ground state) of a hydrogen atom, the most stable configuration. This is because that state's energy is 13.6 electron volts (eV) lower than when the two particles separated by an infinite distance. The dissipation in this system takes the form of spontaneous emission of electromagnetic radiation, which increases the entropy of the surroundings.

What makes an electron jump down to lower energy level?

Now this principle is followed in QM too, and the discreteness of the allowed orbitals is a direct consequence of QM.

The discreteness of the allowed orbitals is a consequence of quantum mechanics, which was conceived precisely to explain this observation, among other things. The discreteness of the orbitals has nothing to do with "stability of the particle" (however, with stability in time, see below), they are simply the only states that appear as solutions to the time-independent Schrödinger equation.

Electron as a standing wave and its stability

If you follow the principle, and the discreteness of the allowed orbitals, then your question "Why are excited states unstable" becomes a direct consequence of these fundamental principles that governs all known atoms in our universe.