It is not that a system follows Le Chatelier's Principle, as it is not a law such as Newton's laws of motion, but rather that this is a shorthand way of working out what happens; a 'rule of thumb' if you wish.
If a system is in stable equilibrium it must have the lowest energy possible under the prevailing conditions. If now a perturbation is made, say by changing temperature or pressure, the system is no longer at equilibrium because some energy has been added or removed. It will now change to lower its energy and again reach equilibrium under the new conditions. Le Chatelier's Principle is then a statement of this.
For example, suppose that as a result of molecular motion, a small part, A, of a larger volume of a gas spontaneously increases by $\Delta V$. Thus the pressure decreases, $\Delta p < 0$, and so the net force on A exerted by the surroundings is in a direction that tends to restore equilibrium.