I hope you're comfortable with that many permissions, I think any takers should think again. Just saying..
The googlevideo.com stuff is required because Youtube serves some content from that domain
Access your data from youtube ... basically the plugin has to parse the page and enumerate the video formats and audio formats and indirectly would be able to "touch" user info
Exchange messages with other programs ... it's because the plugin can download video and audio separately (as Youtube downloads them in browser), for example WEBM for video and Opus for audio, and after they both download, it uses ffmpeg.exe (you can select path to it in the extension settings) to mux these two into a MKV container, or MP4 (if you chose h264 and AAC or MP3 for example)
Download files and read download history etc ... Display notifications... video and audio files downloads are started and can go on for minutes, the extension simply polls the download manager to see when the file downloads are complete in order to know when to start muxing video and audio into a container using ffmpeg (if you chose to mux the two parts). So it needs to access download history to be aware of download status of each stream.
Access browser tabs ... well it has to parse the content of your youtube page to detect the youtube video ID and enumerate available resolutions and all that.
You can find the extension in your profile once you install it, on my computer it's {f73df109-8fb4-453e-8373-f59e61ca4da3}.xpi
It's a plain zip file, so you can simply rename the extension to zip (or just use a software like 7zip which ignores extension) and extract the contents and then you can check the javascript code inside the extension for any access to external sites (to see if it sends data somewhere or not)