How does a generator generate electricity in microscopic terms of the electrons flowing?

In a generator, the free electrons in a rotating coil of wire are pushed along the wire as they move through the field of an external magnet (usually an electromagnet). In an alternator, the electrons are pushed around the loops of a stationary coil by the emf produced by the changing field from a rotating magnet. The alternator (which supplies an AC voltage prior to rectification) has the advantage that much less power passes to the rotator through sliding contacts which can arc and wear. Note that the electrons don't "come from" anywhere; they are just pushed around a conductive circuit.