I can sniff a lot of familiarity with the ''HOW'' answers from what I interpret about you from your post, so I'll only focus on the objective point - ''WHY''.
It turns out that it is possible to meaningfully describe nature by postulating that any object would tend to be in the minimum energy state possible under a given set of physical conditions. So, first we need an understanding of what these minimum energy configurations are - for which we treat the HOWS- (Schrodinger eqn etc.). But once we know what they are, the question is - how do they arrange themselves within these structures, which gets a common sense answer from the <PERSON> principle, which is once again the reiteration of the same idea.
But what is special about this idea becomes clear if you start considering alternatives. Suppose, this wasn't the case, and we chose the stark opposite alternative - every object tended to occupy the most energetic state available (like some bouncing ball collisions in some video games), we would have a really tough time describing nature. For example, we wouldn't be able to explain why any system reaches an equilibrium at all, since e.g. it would be more favorable for an object once set in motion, to keep moving towards an unbounded maximum energy. Now, infinity isn't a number by definition, it reflects an unbounded maximum, so an ''inverted-scale'' description, with infinity in place of 0, is horribly inappropriate. e.g. zero is unique on the number line, but the functions $x$ and $x^2$ both increase indefinitely as we increase $x$. 781
So, ''unbounded from above'' won't be a unique choice, and our description of nature won't be coherent. Anyways, it does so turn out from observational experience that things around us behave as if the underlying principle concerned a minimum, rather than a maximum. So, our postulate seems validated by nature.
Now, to specifically address the question of electronic arrangement, in e.g. <PERSON> atom, which would make a simple example, introducing the quantization conditions (as you are probably aware of, from your question.) Imagine it this way - if electron is getting attracted to the positively charged nucleus, it will tend to bump into it. However, it doesn't, because of its orbital angular momentum (let us neglect spin for the moment), which will cause it to rotate around the nucleus at some orbital radius, owing to a balance between the centripetal force and this attraction. However, while the electrostatic attraction is a continuous function of $r$, falling off as $1/r^2$, with the quantization condition, the angular momentum can't vary continuously, it grows in discrete units. Thus, the balance condition would now imply that all $r$ aren't allowed. You reach a balance at some fixed values of $r$, which define for you the shell locations (Of course the same condition also gives you the permissible energies.)
Now, here's the point (and here's how it relates to my -2 voted first paragraph): once you know the permitted energies, you need some filling up principle, and it most convenient to fill them up using our guiding principle of lowest energy first, rather than the other way round. If we filled them the other way round, we can never explain why there should be a hydrogen atom at all, since the very first electron would be sitting infinitely far away from the nucleus and would behave like an ionized, free electron. 969