Although they are certainly not claiming victory yet, their theoretical work does certainly give one the impression that they are getting closer to understanding high temperature superconductivity.
I found the following two paragraphs from this link particularly informative at explaining how high temperature superconductivity works although, I have to admit, I had to read it very slowly and carefully several times over before I eventually got it!
"....In all known types of high-Tc superconductors—copper-based (cuprate), iron-based, and so-called heavy fermion compounds—superconductivity emerges from the "extinction" of antiferromagnetism, the ordered arrangement of electrons on adjacent atoms having anti-aligned spin directions. Electrons arrayed like tiny magnets in this alternating spin pattern are at their lowest energy state, but this antiferromagnetic order is not beneficial to superconductivity.
However if the interactions between electrons that cause antiferromagnetic order can be maintained while the actual order itself is prevented, then superconductivity can appear. "In this situation, whenever one electron approaches another electron, it tries to anti-align its magnetic state," Davis said. Even if the electrons never achieve antiferromagnetic order, these antiferromagnetic interactions exert the dominant influence on the behavior of the material. "This antiferromagnetic influence is universal across all these types of materials"
I will gladly explain the meaning of that above in much more plain-English layperson terms on request of anyone here. I think a layperson could easily be forgiven for not understanding what they are talking about when the talk about "spin pattern" and also forgiven in thinking (just like I did at first I admit ) the most critical bit I highlighted above in bold sounds like a contradiction even though it is not! I think they could have explained that a bit better than that.