What do massive photons have to do with superconductivity?

Superconductors don't strictly exclude interior magnetic fields. Over a short distance, called a screening length, external magnetic fields can penetrate into a superconductor. This length can be discussed in terms of an effective photon mass, borrowing concepts from particle physics. The heavier a virtual particle, the shorter the distance it can travel before annihilation. The W and Z bosons are very massive and are carriers of extremely short range forces. The longer-range a force, the lighter its force-carrying boson.

Photons are massless, and therefore may carry forces over infinite distances. If describing superconductivity using photons as force carriers that enforce the electromagnetic field condition inside the superconductor, then their finite action distance requires those particles to have mass. This is not to say that there are literally massive photons inside a superconductor, but such a description is possible.

A more detailed treatment is in J.G. Hey, 2004, section 19.2.