Untar anywhere you like and run it from there. I recommend the home directory. You don’t even have to become root. This copy of Firefox should also update itself automatically.
It would be fantastic if this were more like .app bundles on macOS. It’s a special, self-contained folder that GUIs treat as the application binary (double click to launch, etc) but that power users can open as a folder or navigate inside of using a shell.
Then again, I though AppImage was supposed to be something like this?