LLMs choose words based on probabilities, i.e. given the word "blue", it will have a list of words and probabilities that those words should follow "blue". So "sky" would be a high probability, "car" might also be quite high, as well as a long list of other words. The LLM chooses the words not by selecting whatever has the highest probability, but with a degree of randomness. This has been found to make the text sound more natural.

To watermark, you essentially make this randomness happen in a predefined way, at least for cases where many different words could fit. So (to use a flawed example), you might make it so that "blue" is followed by "car" rather than "sky". You do this throughout the text, and in a way that doesn't affect the meaning of the text. It is then possible to write a simple algorithm to detect whether this text was written by an AI, because of the probability of different words appearing in particular sequences. Because its spread throughout the text, it's quite difficult to remove the watermark completely (although not impossible).

Here's an article that explains it better than I can: https://www.kdnuggets.com/2023/03/watermarking-help-mitigate-potential-risks-llms.html