There is only one greyed-out option in the game settings, which is my default installed version (English). How do I change this?
Go to your Origin interface, right click Fifa 16 and choose 'view game details'.
Then click on the 'Game Properties' link and select a new language from the drop down. When you click OK, it will install the new language pack.
You'll have to select the new language from inside the game after it has finished installing. Both the new language and the old language should be available now.