When a magnet is moved into and out of the solenoid, magnetic flux is being cut by the coil.
The cutting of magnetic flux by the wire coil induces an e.m.f in the wire.
When the solenoid is connected to a closed circuit, the induced current will flow through the circuit.
The production of electric current by changing the magnetic field is called electromagnetic induction.
Current/emf is induced only when there is relative motion between the magnetic field and the conductor.
The direction of the induced current and the magnitude of the induced e.m.f due to the cutting of the magnetic flux can be determined from Lenz’s Law and Faraday’s Law.