Trying to learn physics and I'm stuck here. Any help appreciated.

I understand that in the Lagrangian the first component is the kinetic energy, and the second is the potential energy, and together they give the you the trajectory of the object

$L = 1/2 m\dot{x}^2 - V$

The integral of this with respect to time gives the action of the object

$A = \int_{t_0}^{t_1} 1/2 m\dot{x}^2 - V \,dt$

Now I'm supposed to find the stationary point of A using the Euler-Lagrange equation

$\frac{d}{dt} \frac{\partial{L}}{\partial{\dot{x}}} - \frac{\partial{L}}{\partial{x}} = 0$

This is where I get stuck. Why am I applying this? And to what? And what does the stationary point give? Apparently you can derive F = ma from this but I'm confused as to how and why.