Training a simple classifier on internal activations
Probing takes a different angle on the same underlying question: instead of trying to decompose a layer's activations into interpretable features, you ask a narrower, more targeted question, does this network's internal activations linearly represent a specific piece of information. To find out, a researcher trains a simple classifier, often just a linear model, to predict that piece of information directly from the network's internal activations at some layer, while the original network's weights stay frozen.
If a simple linear classifier can reliably recover the information, for instance predicting whether a sentence is grammatically correct, or whether a board position in a game is winning, from a model's internal activations, that's evidence the model is representing that information in a fairly direct, linearly accessible way somewhere in its processing. If even a well-trained probe can't recover it, that's evidence the information either isn't represented at that layer or isn't represented in a form a linear probe can find.