> But is there an intuitive explanation for why it works?
Here's the first great realization about energy:
**Kinetic energy** measures how much is actually happening. **Potential energy** measures how much could be happening, but isn't. The sum of these two, called **energy** is conserved: when more is happening, energy moves from potential to kinetic form.
But this isn't enough to derive laws of physics For that we need the second great realization:
It's also important to think about the difference: kinetic energy minus potential energy. This is called the **Lagrangian**. It measures how much is happening, minus how much could be but isn't. If we integrate this over time we get the **action**, which is very well named. It's the total amount that happened over some interval of time, minus the amount that could have happened but didn't. Nature tries to minimize this. More precisely: if we fix initial and final positions of some collection of objects at the beginning and end of some interval of time, Nature will take the path from initial to final positions that minimizes the action.
A wonderful fact is that this principle, the principle of least action, _implies_ conservation of energy.
I'm not sure what it would mean for you to have an intuitive explanation of "why this works". For that, you need to have some pre-established concept of what it means for it to "work". If for example you already believe in Newton's $F = m a$, then maybe deriving that principle from the principle of least action would be satisfying. You can't derive the principle of least action from $F = m a$ because not all forces are allowed by the principle of least action. In that sense the principle of least action is "deeper". But you can see exactly which forces are allowed, and why.
I'm very slowly writing a book on this stuff, and you can see a draft here:
• John Baez, Blair Smith and Derek Wise, [Lectures on Classical Mechanics](http://math.ucr.edu/home/baez/classical/#lagrangian).
There's a bunch of talk near the beginning, and it already includes some of what I just said, and a lot of other stuff, like a historical introduction to the principle of least action, and how it emerged from the simpler "principle of least potential" in classical _statics_.
Someday I should include a better answer to your question! But it would help to know what your question actually means - that is, what could count as a satisfying answer. There is, of course, a limit on how much we can explain why the Universe has to be the way it is. This limit may change with time, as we gain deeper understandings, but there's always some limit.