[zinclozenge]

Unfortunately, it's not an intuitive idea and it's deeply rooted in the history and development of classical mechanics.

One way of looking at the Lagrangian is via the Legendre transform from the Hamiltonian, which in my opinion is highly intuitive since it's total energy, H = T + V. I'm not going to go through the whole explanation of the Legendre transform. Instead, I want to point out that in the Hamiltonian formulation, you can think of the independent variable as momentum, mv. When you transform to the Lagrangian, you're making, among other things, a change of variable to the velocity, v. If you have a velocity independent potential, then it only really matters in the kinetic energy, and the Lagrangian picks up a sign change due to the transform.

So in the end, it's recasting an intuitive idea into a more mathematically tractable form. The action starts making more sense in the path integral formulation of quantum mechanics.