The tutorial which EA linked previously still has the wrong instructions. Download the Origin client from www.origin.com/download. Open it and go to the MENU BAR (at the very top next to the picture of the Apple), click on > Games > Redeem Product Code...
After I entered my code, I had to restart the origin client (which felt like an unnecessary part of this whole thing) to get my game to show up.
Hope this helps folks in the future, the instructions that come with the disc and on the website are totally out-of-date. Cheers!
