I think of the Lagrangian, what we integate to get the "action", as some sort of energy related function. I don't really attribute much meaning to it other than the fact that minimizing it implies the equations of motion, which are something we can phyiscally grasp.

For a particle in one dimension,

L = L(x(t),v(t))

The solution to the minima is where the "gradient" of L with respect to x and v is zero. However, position x and velocity v are not independent, so that "gradient = 0" equation implies:

dL/dx = d/dt(dL/dv)

- You can define dL/dv is the generalized momentum. - You can think of dL/dx as a force.

This gives you newtons equation, but you can say you derived it.

F = d/dt(p)

Granted, we didn't really start from a more fundamental place. But then this starts to make more sense when you realize the world is governed by quantum mechanics. And this least action principal results from the fact that, in the classical physics regime, the only part of the "trajectory" (wave function) that gives a meaningful contribution is the part along with minima of the lagrangian.