In order to redeem the code follow the instruction below:

Go to http://www.thesims3.com and make an account if you do not already have one. If you already have an account, skip to step 2. If you already have an Origin account, please use the same email and password to register on http://www.thesims3.com.
Once you are a registered member of http://www.thesims3.com, login to your account so that you can then register your game codes.
Click on the My Page tab.
Then click the My Account tab.
Click on Register a Game. With the 16 character key click Redeem a Code
Enter the product code for the expansion pack or stuff pack.
Click the green Register button.

If the method above won't work, please follow the instruction below:
 

If you have not done so already, create your free Origin/EA account on www.origin.com.
Log in to the Sims 3 community website using your Origin email and password. If you are logging in for the first time, use your provided key for the Sims 3 base game during the registration process.
To redeem the games on your Sims 3 account select "My Page" -> "My Account" -> "Register a Game"
Enter your 20-character key for The Sims 3, Late Night Expansion, or High End Loft Stuff. (note, if you entered the The Sims 3 base game code in step 2, you don’t need to re-enter it here)
That's it! The Games will automatically be added to your Origin account, and can be accessed and downloaded through the Origin client.