Lenz’s Law bypasses the mathematics and provides an intuitive short-cut to figuring out the direction of the induced emf: the induced emf is always in the direction that opposes (or tries to oppose) the cause of the induction.
Take for example when emf is induced by an approaching magnet. We can see the increasing leftward magnetic flux (of the coil) as the change that is causing the induction. To oppose this change, the coil should (try to) produce a rightward magnetic flux. This requires an anticlockwise current in the coil. This allows us to deduce that the induced emf (and thus current) in the coil is anticlockwise.
Alternatively, one may see the approaching magnet as the cause of induction. To oppose this change, the coil should (try to) repel the magnet away. This requires the coil to present a north pole to the approaching magnet, which requires an anticlockwise current in the coil. So we arrive at the same conclusion as before.
How about the case when emf is induced by a retreating magnet? If we see the decreasing leftward magnetic flux (of the coil) as the change that causes the induction, then the coil should (try to) produce a leftward magnetic flux to oppose this change. That requires a clockwise current in the coil. So we can deduce that the induced emf (and thus current) in the coil is clockwise.
Alternatively, we can see the magnet moving away as the cause of induction. To oppose this change, the coil should (try to) present a south pole to the retreating magnet to keep it from leaving. This requires a clockwise current in the coil. So we arrive at the same conclusion as before.
Principle of Conservation of Energy
Some astute students may recognize Lenz’s Law as a manifestation of the Principle of Conservation of Energy. Note that an approaching magnet produces electricity in the coil. The creation of electrical energy must be at the expense of the kinetic energy of the magnet. If the magnet is being attracted instead of being repelled (meaning the change is reinforced instead of being opposed), the magnet will be gaining KE even as it is inducing electrical energy. That’s a clear violation of PCOE!
For the situation of a retreating magnet, the magnet ought to be experiencing an attractive magnetic force. Only then can the creation of electrical energy in the coil be accounted for by a corresponding decrease in the KE of the magnet.
In practice, the magnet can be kept going at constant speed by an external force acting in the direction of the magnet’s velocity (leftward for the approaching magnet, rightward for the retreating magnet). In this case, the creation of electrical energy is accounted for by the (positive) work done by the external force on the magnet.