The piece of paper with the picture on it just gives instructions.

Entering your serial code:

1. Visit the FFXIV Mog Station.
2. Log in with your S-E account (the same one you use to play).
3. Select Your Account> Enter Item Code as mentioned above.
4. Enter your unique code (from the cardboard slip on the obi card). It has to be entered in uppercase if it's printed in uppercase, otherwise you get an unhelpful error message.

Note: If you were logged in when you did this, you might need to log out and back in again for the moogle mail to arrive in-game.