I agree with the recommendation to update Xandros to a current distribution. I personally would simply recycle the EEE and get newer/better hardware, but if there are budget restrictions, AntiX is a very nice distro for low-end hardware.
To answer your specific question, a tar.bz2 file is simply an archive (like .zip in windows) so you extract it using whatever archive tool Xandros comes with (I don't know because I've never used Xandros) then, once the folder is extracted, navigate to it with your File Manager and double-click the Firefox binary inside (like .exe in windows). However, as mentioned above, it is possible that Xandros is too old to satisfy the requirements for the current Firefox... I really don't know. It is like asking "how do I bake a pie with rotten cherries?"
Perhaps it can be done, but the pie will taste better if you update to fresh cherries. Fortunately, Linux is like free cherries, so the only cost is your time/effort.