Here's the thing, let's say I gave you an alleged answer to the question of "why". For concreteness, let's say my answer is: "Because of the Principle of Stationary Action".
Now, my question to you is this: would this answer satisfy your question of "why" or would it be the case that your curiosity would lead you ask me: "But why is nature such that the Principle of Stationary Action holds?".
It's possible that I might even give you an alleged answer to that. But, if I did, the "Principle" of Stationary Action would no longer be a principle.
The point is this: if something is fundamental or is a principle, asking "why" presumes that it isn't fundamental or a principle. For, if there were a genuine answer to "why", that answer would be the fundamental or principle instead.
At some point, the only "answer" is "It just is".