Chrome and Chrome-based Browsers
If you are running a Chrome-based browser (which is pretty much anything nowadays besides Safari and Firefox), you need to perform extra steps to install the extension.
Long story short, the Chrome Web Store is run by Google who owns YouTube and will kick out any extension that allows downloading (or helps downloading) content from YouTube. As all of the browsers want you to use the Chrome Web Store to install the extensions, you need to install it manually.
Please note that there are unofficial Downie extensions on the Chrome Web Store which will not work with YouTube.
To install the extension, click on the Chrome button in Settings > Browser Extensions. On macOS 13 or later, you will get an installation wizard that will do most of the work for you and guide you step by step through the rest. On earlier macOS versions, a blog page with instructions will open.