Great question! In order to get a reward for referring a friend, your friend will need to provide your username or referral code when telling us how they heard about Animal Jam. This will award you with 1500 gems. In order to get the reward of 15 Diamonds and a 1 week membership, your friend will need to become a member of Animal Jam after having used your username as the person who referred them to the game.
If you haven't received your rewards please log out and back in, it can take up to 24 hours for them to arrive.