Explain the Lenz Law

An induced current always flows in a direction so that its electromagnetic field opposes the change in magnetic flux producing it. This is embodied in the negative sign on the right side of Faraday's Law. If this were not true, if the induced current could be in the same direction as the change in flux causing it, then we would have a system which would make the flux bigger and bigger with no end in sight - this is not likely to happen if energy is conserved.

The magnet moved into the coil, the induced current in the coil has to make the electromagnet of the coil repel the magnet being moved towards it. So, if the coil shown is wound counter- clockwise when viewed from the right side, the positive induced current would go counterclockwise in the circuit to create an electromagnet, which has a magnetic field opposite the one in the permanent magnet--so it will repel the permanent magnet.

Try example of the magnet moved out of the coil. Which direction is the induced current in the circuit?

Answer - No peeking until you have tried on your own!  Now that you have given it a try, check your answer against this. The magnet is being pulled out of the coil so that the electromagnet caused by the induced current must be in the direction that attracts the permanent magnet. This means that the induced current must flow from the left to the right side of the solenoid, and the right hand curls from the back to the front of the solenoid to make an electromagnet with its magnetic field pointing to the right, the direction of your right thumb. The induced current flows clockwise in the circuit.