Below are download links for the generated transcripts used by the "Custom" option on the search page. YouTube's generated transcripts have been removed from these downloads, they're not as accurate anyway.
With these files you will be able to search for quotes offline, as an alternative to visiting this site.
If new videos are archived on the site, these downloads will be updated.
The SRT files contains captions for a video. They're plaintext so can be opened and easily read in notepad, but you can also open a video and the matching SRT file in a media player like VLC for captions.
The DB file is a SQLite database. Open the "subtitles" table and search through the "text" column for quotes. Join the "id_transcript" column with the "id" column in "transcripts" to find the video the quote came from.