It uses an alternative backend where videos pre-removal had their dislike counters archived, and anything post-removal uses dislike data from users with the extension to crowdsource a likely dislike counter.

(in theory, if 5% of viewers of a video are people that have the extension, and only 10% of those viewers hit the dislike button, chances are that's a good sample size to assume that this video should have a 10% dislike ratio, but in practice it's probably a bit more complicated/weighted)