Does anyone have more information about the finding itself? This article seems to be mostly a story about how "there was this mystery, and this scientist solved it with math".
Here's the preprint of the article in Phys. Rev. Lett. that this is a press release for. Note that it's been on arXiv for almost a year now.
Also note PRLs are brief and not very detailed, so it should be readable for physicists not working in this particular field.
Yaida found that for mean-field models of spins with quenched impurities in an
external magnetic field, in three dimensions, the renormalization group flow admits a fixed point at the two-loop level of calculation. It was long known that a fixed point exists in dimensions d > 6, but earlier one-loop calculations didn't find one for d=3, so it was thought that the nice results from these mean-field models had no physical relevance. This new result indicates that we can indeed use these models to understand physical disordered systems.
Can you give a suggestion for understanding the connection between magnetic fields and glass? Do we need to understand quantum physics in order to properly appreciate this connection?
Physicists extend "glass" into "disordered systems". A system can be ordered or disordered in different way, e.g. order (crystal) or disorder (glass) in the positions of atoms, or the simpler case of ordered or disordered spins of atoms (either "up" or "down"). The same phenomena exist in both fields, e.g. glass transitions, so we prefer to study the simpler "implementation", namely spins in a magnetic field.
Computer analogy: we're trying to reverse-engineer an alien ASIC, which can operate either on an array of tuples of three reals, or on an array of bools. It works approximately the same in both cases. The latter case is much easier to analyze, so we prefer to do that.
Analogously Neural Network learning, i.e. learning modeled as a phase transition between disorder and order:
“... studies the connection between the highly non-convex loss function of a simple model of the fully-connected feed-forward neural network and the Hamiltonian of the spherical spin-glass model”
from "The Loss Surfaces of Multilayer Networks"
Choromanska, Henaff, Mathieu, Arous, & LeCun [1]
The other comment also led me to https://en.wikipedia.org/wiki/Spin_glass, which I didn't know about at all (I'd heard the term once but had no context for it).
"Annual Review of $TOPIC"-journal papers are usually very good. Note also that the lead author on the review you linked is the advisor of the postdoc that did the work described in TFA.
