Magnetic flux through a loop is the flux density multiplied by the area that the field passes through at right angles. An EMF is induced in a circuit whenever the flux linking it changes: the field may change, the area may change, or the coil may turn. Faraday's law says that the size of the EMF equals the rate of change of flux linkage. A steady flux, however large, induces nothing.
Lenz's law gives the direction: the induced current flows so that its own magnetic effect opposes the change that causes it. This is the conservation of energy. Work must be done against the opposing force, and that work becomes electrical energy.