How does Lenz's Law illustrate the concept that "you can't get
something for nothing" as far as energy is concerned?

When there is a change in magnetic flux linkage through a loop of wire, an electromotive force is induced in the loop, according to the Faraday-Newmann-Lenz Law:

where

N is the number of turns in the loop

is the change in magnetic flux through the loop

is the time elapsed

The negative sign in the formula represents Lenz's Law, and tells us about the direction of the electromotive force.

In fact, the negative sign means that the direction of the induced emf is such that to oppose to the change in the magnetic flux that originated the induced emf.

This is a consequence of the law of conservation of energy: no energy can be created out of nowhere. In fact, when the emf is induced in the loop, electrical energy appears in the circuit; however, this electric energy cannot come out of nowhere. Instead, it is just "created" from the transformation of some other form of energy (for instance, the mechanical energy that is used to move the loop in the magnetic field, and changing its magnetic flux).

The negative sign in Lenz's Law tells exactly this: the direction of the induced emf is such that it opposes the initial change in magnetic flux that generated the induced emf, so that overall the total energy is conserved.