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This is due to Lorentz Force.
There are plenty of free electrons in a conductor. When there is a relative motion between the conductor and the magnetic field (either the conductor or the magnet is moving, and the other is stationary), interaction occurs between the charges on the free electrons, which associated with them an electric field, and the magnetic filed. Such interaction between electric and magnetic fields produces a force, commonly called the Lorentz force, that drives the electrons to move. This phenomenon is known as electromagnetic induction. The movement of electrons therefore constitutes an electric current.