As of this posting, their FAQ explains everything you need to do:
WHO IS ELIGIBLE FOR THE FREE BIRTHDAY BURGER?
We love celebrating your birthday! The Free Birthday Burger reward is available for members who provide their email, a validated phone number, sign up for promotional marketing through SMS or Email, and also provide their Date of Birth. Once it's your birthday month, we will send you a reminder to come get your free Birthday Burger.
HOW DOES MY BIRTHDAY BURGER REWARD WORK?
If you are eligible for the Free Birthday Burger reward (see above), it will be automatically activated on your account the first day of your birthday month, and may be redeemed once during that month using when you apply your Royalty account to the purchase. You will receive an e-mail notification or text message once the reward is available on your account.
The Birthday Burger reward includes any of our fire-grilled Gourmet Burgers and Finest Burgers but does not include add-ons. The Birthday 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 per account.