What makes an electron jump down to lower energy level?

The falling to the lower level for electrions in an atom. is due to sponaneous emission. One can think of it as being due to the infuence on the excited state of zero-point-energy vacuum fluctuations in the electromagetic field. The necessary existence of spontaneous emission was deduced in form of A and B coefficients by Einstein in 1916 --- long before the idea of quantum fields or even zero point energy was about.

Before going to the quantized atomic levels, consider that in classical physics there is the Minimum total potential energy principle

A rolling ball will end up stationary at the bottom of a hill, the point of minimum potential energy. The reason is that as it rolls downward under the influence of gravity, friction produced by its motion transfers energy in the form of heat of the surroundings with an attendant increase in entropy.

Principles have the strength of axioms in the theories of physics, they are imposed because then the mathematical theories fit the observations and are predictive of new ones..

This principle is carried over in the case where quantization is imposed, so that the electrons around the nucleus instead of following the principle of minimum energy and ending up neutralizing the nucleus, they stay in stable orbits allowing for atoms and solids to form. If a lower orbital is empty, the principle forces the decay to that orbital, within the probabilistic rules of quantum mechanics, as long as there is a way for the extra energy to be dissipated, and that is the emission of a photon.

from the link

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.

Why does an electron drop to a lower energy level? Why questions are often outside the scope of physics. Physics says how the universe behaves, not why it does so.

But sometimes you can say why. Sometimes a law is derived from a simpler law. Why does the complex law work? Because it is a consequence of the simpler law. It still might leave you wondering why the simpler law is true. Or you might be so used to the simpler law that it feels satisfactory. The simplest laws are justified by agreement with experiments.

In this case, we can start with a classical electron being attracted to a proton. Protons exert an attractive force on electrons. This is the simpler law that I cannot explain further. (Quantum mechanical explanations for the attraction do exist. But you will get to a point where you cannot explain why quantum mechanics works. It just agrees with experiments.)

The proton has an electric field around it. You can define more complex laws: The forces on an electron depend only on its position.

$$PE = F cdot d$$

$$KE = 1/2 mv^2. $$

From the properties of forces, you find that $PE + KE$ is conserved.

In this case, you can say an electron tends to move to a lower energy position because forces point toward lower energy positions, and electrons are accelerated in the direction of forces.

The same type of argument can be made for quantum mechanical electrons and protons, but it is harder. It is easier and more natural to work directly with energy. This is why physicists tend to work with energy and think that moving to lower energy is the simplest law.

A quantum mechanical electron is not a point particle. It has no single position, so you can't say that a single value of force is acting on it. It has no trajectory. So you can't integrate $F cdot d$ over the trajectory to get $PE$.

But it does have a wave function. There is a time evolution operator that you can use to calculate the wave function at future times. You can calculate probable position distributions from wave functions. From this you can show that electrons tend to move toward protons.

You can also show that the uncertainty principle keeps them from getting too close. An electron that is very close to a proton has a small uncertainty of position. Therefor it has a high uncertainty of momentum. An electron that may have a high momentum isn't very likely to stay near a proton. Orbitals are stable configurations where there is a balance between these two tendencies to be attracted and move away.

Higher energy orbitals are in general larger than lower energy orbitals. So an electron tends to move toward lower energy orbitals. Classically, it would gain kinetic energy as it does so. Quantum mechanically, it emits the energy in a photon instead.

The behavior of the quantum mechanical electron is governed by quantum mechanical laws. One of the chief laws is the Schrodinger equation. It is a quantum mechanical version of the classical conservation of energy law. So you can say that quantum mechanical electrons move toward lower energy positions for the same reason that a classical electron does.