Copy a URL from a YouTube video, then open up VLC. From the Media menu select Open Network Stream (Ctrl+N) and paste in the URL. Click Play. When the video is playing, go to Tools > Codec Information. There is a box at the bottom called "Source"—copy the URL you find in that box.
Go back to your browser and paste the URL—it's a temporary web address and will expire eventually—in the address bar and start playing the video. When you're viewing the video in the browser, you'll see an option to download the file to keep locally.
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