[krukah]

It's a great question that, as far as I can reason, has no answer. Newtonian vibes that are familiar to us will only take you so far, and intuitive interpretations of physical quantities often break down when you try to relate them to the scale, experiences, and stimuli of humans.

Let's take momentum, energy, and charge, things that you probably have a strong "real world feel" for. It's worth noting that our intuition for these quantities is actually pretty far-removed from their mathematical origin. Maybe you consider these as different loosely related quantities that pop up in different loosely related calculations, which is a useful and powerful mental model. Momentum is a thing that..."gives velocity to inertial bodies". Energy is a thing...that "does work". Charge is a thing that..."causes forces in the presence of an electric field". If you try to define the terms within each definition, you'll find yourself in some circular definitions, and it'll become unclear which definition, if any, is "most fundamental".

But these quantities are actually quite similar in the sense that they can all be defined in terms of action! Specifically, these are quantities that are conserved because there exists some nice symmetries in the Lagrangian (roughly speaking, a derivative of action). So our intuitive definitions of these things are really just less generalized/more specific understanding of structure that is emergent from action.

Can we look at a physical system and say "oh this one's got a lot of action" or "nature's doing a great job of minimizing the action over here"? No, but we can look at a physical system and say "wow, everything that's happening in here lines up with what I'd observe if this little quantity I defined just so happened to be minimized"

I think no matter how many Lagrangians we integrate or variational calculations we perform, we'll probably never gain a better intuition for action beyond "The Thing That Explains A Lot Of Seemingly Unrelated Physics When It's Minimized." To me, it's both deeply unsatisfying for its abstract and unintuitive nature, but also deeply profound for its universal explanatory power.

tldr; when it comes to action, reject real world feels and embrace mathematical structure.