They're trying to explain the phenomenon, where during the training of a deep neural network, sometimes it will spend a long time with suboptimal parameters and then suddenly improve a lot very quickly, which has been termed "grokking." To do so, they study deep linear networks instead, which are easier to analyze. In that setting, it's possible to show that for more than two layers, there are two solutions for each feature, one where the network handles the feature correctly and one where it ignores it completely. However, that second solution has worse loss, so it is only metastable: if a path can be found to a region with better loss, the model will go there instead and "grok" the feature.

This path is provided by noisy parameter updates: the model learns from random samples of the training data, so every time it changes in a slightly different direction. Given enough time, those directions can randomly add up to a lucky escape path from the metastable solution. This is analogous to thermodynamics, where you can have a particle randomly bouncing around until it encounters another particle it can combine with in a lower-energy state, and the higher the temperature, the more quickly it happens. They empirically measure the relationship between escape time and temperature and find that its form agrees with the thermodynamic explanation, although their theoretical approximation of the energy barrier differs a lot from what the measurements imply, which they attribute to additional corrections necessary to make the approximation exact.

(In other words, it's impossible to summarize this in one sentence without first explaining a bunch of background information.)