You will find your Player ID in this section. The Player ID is what people use to add you to their Friends List, not your player name (the name you entered at the end of the tutorial). You can add your Friends' Player IDs here.
On the top right corner, you will see a Referral button. You can add one friend as a Referral. When you reach Rank 20, you and your friend will both get 50 Diamonds. Once you've added a friend, that button will disappear.
But others can also add you as a Referral, and when they've reached Rank 20, you will both get 50 Diamonds each.
This will only work for the first 5 Referrals, meaning, only 5 other people can add you as a Referral.