The fact that parts of human linguistic concept-space can be encoded in a high dimensional space of floating point numbers, and that a particular sequence of matrix multiplications can leverage that to perform basic reasoning tasks is surprising and interesting and useful.
But we know everything about how how it is trained and how it is invoked.
In fact, because it's only "state" aside from its parameters is whatever its context window, current LLMs have the interesting property that if you invoke them recursively, all of their "thoughts" are human readable. This is in fact a delightful property for anyone worried about AI safety: our best AIs currently produce a readable transcript of their "mental" processes in English.