It fetches the video's public watch page and reads the keywords meta tag YouTube still embeds in the source for every video, then de-duplicates the list and shows it in the order the uploader entered it. Results are cached, so re-checking the same video is instant.
Using them well
Don't copy another channel's tags wholesale — YouTube's own guidance says tags have minimal ranking effect, and identical tag lists on unrelated videos can look like spam. Use them as a vocabulary check: if the top-ranking videos for your topic consistently use a phrase you don't, that phrase probably belongs in your title or description. The Tag Generator builds a list from your own title if you're starting from scratch.