That guide isn't very specific about where to extract the .tar.xz file to. It just says: "When the download is finished, expand the Arduino tar.xz package into a folder where the Arduino Software (IDE) will run."
The best place is probably under /opt/. If you want it there, move the .tar.xz file into /opt/ and extract it there. You may have to do this using sudo or as root and the same may apply to running the
./install.sh command but try each as your own user ID first.
However, if you just want it installed locally in your own Home folder, I wouldn't put it in a folder on the Desktop. I'd create a folder under Home called something like ArduinoIDE (don't use just Arduino). Move the .tar.xz file into this folder and extract it there.
Once installation is complete, you can delete the .tar.xz file, if you wish.
If you've installed a version of the Arduino IDE using your distribution's package manager, you should uninstall that first.