Youtube to MP3 Downloader provides yt2mp3app:// protocol which allows third-party applications to invoke it from outside and send URL parameter while doing it, so it will start application with that URL pre-filled.
This extension leverages that protocol; it renders download button on YouTube video page, which when clicked uses mentioned protocol to start the application with URL from current video.
This extension isn't yet published in Google Chrome Store and at the moment is only available through loading it as unpacked extension. First download this repository and then consult Chrome's documentation on loading unpacked extension