If you point Firefox to http://www.mono-project.com/Moonlight/, get to the download page and click on download after selecting the 64bit architecture, it will automatically install the extension.
Chrome/Chromium/Iron, on the other hand, will prevent the .crx extension to be installed automatically.
In this case, after you have downloaded the .crx file from the Moonlight download page (using Chrome), go to the Tools | Extensions configuration page, and then drag the .crx file onto the Chrome extensions page. It will ask for your permission to install the extension.
The .xpi extension package "is simply a ZIP file containing the extension files", so you can open it just like any other .zip file. Note that you might need to change the extension of the file to .zip, depending on the method you use to extract the files.
One simple way to extract the contents of a downloaded .xpi, is with the use Midnight Commander, which can be easily installed:
sudo apt-get update && sudo apt-get install mc
Navigate to the folder where the file has been downloaded and it can be opened like a folder and its contents copied out with F5.