Even though the historical order is the other way around, it is helpful to start from wave/quantum mechanics and arrive at classical mechanics in the limit that the wave length of the particle goes to zero. Mathematically, that limit is the <A HREF="https://en.wikipedia.org/wiki/Stationary_phase_approximation">stationary phase approximation,</A> meaning that classical trajectories are perpendicular to surfaces of constant phase. The phase is the action = integral of Lagrangian $L$, as first realized by Dirac, and as follows immediately by calculating the phase $\phi$ accumulated in a time $T$,
$$\phi=\int_0^T(p\dot{q}-H)\,dt=\int_0^T L(q,\dot{q})\,dt.$$
From this equation you see that the Lagrangian is "kinetic minus potential energy" because it is the difference of $p\dot{q}$ = twice the kinetic energy and the sum of kinetic and potential energy (the Hamiltonian $H$). In this way stationary phase amounts to stationarity of the action.