In a conductor, charges are free to move about. If there is a nonzero electric field inside a conductor, the charges would of course move. The charges continue to move, forming accumulations of positive/negative charge in various regions of the conductor, until they reach a configuration with zero net electric field (external field plus the field generated by the charge accumulations in the conductor), and in that configuration they then stop moving. In practice, these charges accumulate on (or within a "skin depth" of) the surface of the conductor--the charges accumulate on the surface of the conductor in just the right configuration to exactly cancel any field inside the conductor.