Why and how is the supercurrent generated in a superconductor in response to an applied magnetic field? Does it flow only along the surface? If yes, why?
When a magentic field is changing in time, there is an induced electric field from Faraday's law. As a magnet approaches the superconductor, and so the magnetic field is increasing, this induced electric field starts a current flowing in the surface of the superconductor. This induced current produces a magnetic field that exactly cancels the field of the magnet in the bulk of the superconductor, and so effectively pushes the magnet's field out of the superconductor.
The reason that the current is located near (within a penetration depth) of the surface is that a magnetic field within the superconductor causes a rotational current within the superconductor, and this costs energy. There is less energy cost if the current stays close to the surface.
A more detailed, quantitive, discussion requires using Maxwell's equations and either the London or the Landau-Ginzburg equations for the superconductor.