I'm guessing it essentially increases the elastic constants of the material; that is, increases the "spring constants" of the bonds between atoms. As a result, the resonant frequency of acoustic vibrations rises. When these acoustic vibrations are quantized into phonons, the minimum energy of each phonon therefore rises, since it is proportional to this resonant frequency. Since conventional cooper pairs are pairs of electrons bound by a phonon, this increases the binding energy of the pair, and makes it more resilient to being broken up by the thermal energy.

You can have water under high pressure that does not freeze at 0 degrees, as it would at atmospheric pressure. It's because there are phases that water has behave differently under different pressure and temperature. Water behaves very differently at different temperatures and the same goes for Lanthanum (the metal used to create the superhydrate, or whatever its called). At room temperature the metal just does not bond to as many hydrogen atoms as needed to create the superconducting latice (the paper goes of the model that metals that can bond to >6 hyrogen atoms per host molecule are good candidates for room temperature superconductivity) so we have to do it under extreme pressure.

The traditional way that superconductors have been made has been to lower the temperature near absolute zero to get the metals into that a superconducting phase. What these researchers are trying to do is prove a model that suggests that there are metal compounds with superconductivity at near room temperature.

The material forms certain crystal structures at those pressures which were predicted to be superconducting.

Stupid answer from an organic chemist with limited inorganic and material chemistry:

In theory it would compress the material to a quasi crystal structure. This compressed material would allow the electrons to freely jump from atom to atom/molecule to molecule without hindrance or loss.

Note: This is pure guessing!