I don't think we need to explain anything. They know exactly how their system works and will have heard of Apple Pay. Just ask them if restaurant transactions made with registered cards through Apple Pay will be tracked and credited, and if not how you can register your Apple Pay device.
I sent an email to Rewards Network regarding Apple Pay and here was their response (bolding and red font added by me):
Thank you for contacting us.
To earn benefits through our program, your current registered credit card must be swiped at the participating restaurant. Unfortunately, transactions processed through a third party company, such as Apple Pay, would not earn benefits.
If you have any further questions or concerns, please do not hesitate to contact me or call Member Services at 1-888-419-6683.
Sincerely,
Member Services