This principle states that the electron starts filling from the lowest energy up. Why?
The state where the electron is closest to the proton has the minimum potential energy since the two particles attract each other. Once these states are occupied (Pauli Exclusion Principle) the next electron has to "settle" for a position further away from the positive charge as the position closer to the proton is already occupied by an electron. And so on.
This is also the reason why electrons in the ground state of an atom that are promoted to higher energy levels by, for example, a photon typically don't last there for long (has short lifetimes). The reason is that if there is an "open state" of lower energy available where an electron can move closer to the nucleus it will fill that state at its earliest opportunity.
This is a simplistic view to be sure (so I don't know if it qualifies as being the most "fundamental" explanation), but it fits in with a lot of chemical and physical observations.
Is there a physical law such as systems tend to reach a state of minimum potential energy or something?
It's just an observable fact. There are different ways to see these sorts of phenomenon in classical physics with gravity and the electrical force, in thermodynamics,statistical mechanics with energy distribution, and these ideas carried over into quantum mechanics as well. For instance, you can also see this trend with entropy, as entropy tends to spread out energy as evenly as possible between all the available energy states. Therefore if you have one energy state higher than the energy state of nearby objects it will tend to spontanteously lose energy until its energy state matches that of its surroundings. That's just one way of seeing this motif (the electron energy levels is another).
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