Downloading videos from youtube is 'illegal', so to say.

And I quote "You shall not download any Content unless you see a “download” or similar link displayed by YouTube on the Service for that Content. You shall not copy, reproduce, distribute, transmit, broadcast, display, sell, license, or otherwise exploit any Content for any other purposes without the prior written consent of YouTube or the respective licensors of the Content. YouTube and its licensors reserve all rights not expressly granted in and to the Service and the Content." 
        http://www.youtube.com/static?gl=US&template=terms

Are you trying to install the extension on Firefox for Android?

Desktop Firefox extensions do not automatically work in Firefox for Android. It is up to each individual extension author to support new platforms.

YouTube Video & Audio Downloader is compatible with FF 25. https://addons.mozilla.org/en-US/firefox/addon/youtube-video-and-audio-dow/

If it is not working for you then you need to create a new profile: https://support.mozilla.org/en-US/kb/profile-manager-create-and-remove-firefox-profiles#w_creating-a-profile