the meissner effect is a consequence of lenz's law:
the initial relative motion between the magnet and the superconductor sets up the eddy current (according to Lenz's law). remember that there is no resistance, this means 1. theoretically the current would be infinitely large, creating an infinitely large magnetic field, and hence why it is impenetrable 2. the current would remain in the superconductor indefinitely.
as such, the magnet becomes repelled because it has created an infinitely large opposing force.
now consider that this overcoming force has become so large that the magnet moves the other way.
this creates relative motion, but this time in the opposite direction as to before. this creates an eddy current that would reduce the strength of the previous eddy current.
this process would continue until the magnet can be seen to visibly vibrate in mid-air. an analogy is the self-regulating nature of DC motors, with the back EMF, or the slip speed variation in induction AC motors.