In the tar.bz2 archive you downloaded is a directory called firefox. Extract that to where you like.

To run it open a terminal and type <where you put it>/firefox. You can also navigate to it and double click the firefox file with whatever graphical file browser you use.

Official instructions: https://support.mozilla.org/en-US/kb/install-firefox-linux#w_install-firefox-from-mozilla-builds

What Desktop? X11 or Wayland? Try downloading Firefox from Mozilla. Download, unzip, and run firefox-bin from the folder. https://www.mozilla.org/en-US/firefox/all/#product-desktop-release