As of this posting, their FAQ explains everything you need to do:
WHO IS ELIGIBLE FOR THE BIRTHDAY BURGER REWARD?
The Birthday Burger Reward is available for members who have provided their email, verified their phone number, signed up for promotional marketing through text or email, and provided their date of birth. When your birthday month comes around, we’ll send you a reminder to come to Red Robin to get your free Birthday Burger. This reward is redeemable for Dine-in only with a minimum purchase of $4.99.
HOW DOES MY BIRTHDAY BURGER REWARD WORK?
If you are eligible for the Birthday Burger Reward (see above), it will be automatically activated on your account the first day of your birthday month. You will receive an e-mail notification or text message once the reward is available on your account.
Your free Birthday Burger is redeemable once during your birthday month during a Dine-in visit only, and requires a minimum purchase of $4.99. You can choose from any one of our fire-grilled Gourmet Burgers and Finest Burgers but it does not include add-ons. The reward will be automatically loaded to an account 24 hours after registration is completed and it will expire when your birthday month ends. Note that there is only one Birthday Burger Reward per account.