This is a field I find fascinating. It's generally the research field of Machine Learning Interpretability. The BlackboxNLP workshop is one of the main places for investigating this and is a very popular academic workshop
https://blackboxnlp.github.io/One of the most interesting presentations in the last session of the workshop is this talk by David Bau titled "Direct Model Editing and Mechanistic Interpretability". David and his team locate exact information in the model, and edit it. So for example they edit the location of the Eiffel Tower to be in Rome. So whenever the model generates anything involving location (e.g., the view from the top of the tower), it actually describes Rome
Talk: https://www.youtube.com/watch?v=I1ELSZNFeHc
Paper: https://rome.baulab.info/
Follow-up work: https://memit.baulab.info/
There is also work on "Probing" the representation vectors inside the model and investigating what information is encoded at the various layers. One early Transformer Explainability paper (BERT Rediscovers the Classical NLP Pipeline
https://arxiv.org/abs/1905.05950) found that "the model represents the steps of the traditional NLP pipeline in an interpretable and localizable way: POS tagging, parsing, NER, semantic roles, then coreference". Meaning that the representations in the earlier layers encode things like whether a token is a verb or noun, and later layers encode other, higher-level information. I've made an intro to these probing methods here: https://www.youtube.com/watch?v=HJn-OTNLnoE
A lot of applied work doesn't require interpretability and explainability at the moment, but I suspect the interest will continue to increase.