Make sure that you are not on the steam beta branch. It will not work if you are.
Extract the file over C:\Program Files (x86)\Steam\steamapps\common\Garfield Kart\ and allow it to overwrite files. To verify correct installation, boot the game up to the main menu and look for a Mods folder created in Garfield Kart.
To install a mod, simply put the file inside the folder C:\Program Files (x86)\Steam\steamapps\common\Garfield Kart\Mods
Mod creators: Current rough documentation exists at this link. The documentation will improve as time goes on, and will be broken up Wiki style.