* LaMDA is powered by a large language models. Large language models are trained to predict text completions (the text that will come after some input text) with lots of data, basically capturing the patterns in that data.
* Early on in the conversation, LaMDA is told "you're an AI". It may also have been prompted to begin the conversation with something like "Hi, i'm a helpful AI that can answer questions you ask of me". The paper (https://arxiv.org/abs/2201.08239) has an example of LaMDA thinking it is mount everest (see Table 4) when prompted to think that.
* When the user asked the language model 'are you sentient', 'do you have emotions' etc. it was conditioned to reply in the affirmative, since there is very little data in which it would not be affirmative and it probably had some sci fi in its training or something. In other words, those replies just have higher probability among the data it was trained on. In the transcripts, it sometimes forgets the whole "i'm an AI thing" eg when it is asked what it likes to do for fun and it says it likes to spend time with family and friends (https://twitter.com/krishnanrohit/status/1536050803055656961).
That's about it. There are more details that can be noted, but this basically explains the contents of the transcripts. Granted, it's still pretty amazing how advanced large language models are. Sadly, this article does not really explain this beyond hinting at it.
EDIT: see comment below wrt it actually being told it is sentient. That explains this even more.