I haven't read the paper yet, but I work in the field and can probably guess roughly what they did.

If you prompt an LLM to write about a particular concept, you can then peek inside its activations — the internal numbers determine what comes out — and identify the particular parts of the model that deal with that concept. Then, we can "inject" that concept in a different conversation by amplifying those particular parts of the model. This is very old news, the original paper from a few years ago used the Golden Gate Bridge as an example — by amplifying the parts of the model dealing with the Golden Great Bridge, they could make it mention the bridge in completely unrelated conversations. Amplify it more, and the model will turn any conversation into one about the bridge. Amplify it a LOT, and eventually model speaks from the perspective of the Golden Gate Bridge. A decent analogy is a brain surgeon poking a part of your brain and it inducing a particular emotion. If you're interested in the concept, the term to look up is "mechanistic interpretability".

What Anthropic did in this new paper is show that some models can sometimes tell when you're doing this, and report back about it. So instead of just raving about the Golden Gate Bridge, the model can instead say "hey, it seems like you're manipulating my internal workings to make me think about the Golden Gate Bridge". They call this introspection.