When a charged particle is moving in a magnetic field, a force known as the Lorentz force acts on the particle. The magnitude and the direction of the force are given by the following equation:
$$\vec{F} = q(\vec{E} + \vec{v}\times \vec{B})$$
Consider a piece of conducting wire. The conducting wire consists of freely moving electrons which experience a force in the presence of a magnetic field. However, there won't be any force acting on the wire as a whole as the electrons are moving in random directions and the net force adds up to zero.
When you move the wire in a particular direction, the vector sum of the direction of motion of the electrons in the conductor adds up to give a finite value. Hence, when you sum the forces up vectorially, you get a net force.
This causes the electrons to move in a particular direction which in turn results in a separation of charge as shown in the diagram.

This establishes an electric field along the wire which prevents recombination of the charges. Hence, there exists an E.M.F along the wire.
When you complete the circuit by joining wires at the ends of the rod, the electric field which was preventing the charges from recombining drives a current in the circuit.