They email you a link that you can use to access your certificate. The page it links to has your ceritificate which you would then print. My guess is that this is so that it can only be printed once.
It's that printed certificate that you take to the restaurant. There is a phone number on it that the restaurant management would use to authenticate the thing.
I had no worries about using my credit card. It's a very professional site.
Protip: if you have a second email address you use for non-important things, you might want to use that. I'm glad I did. After I signed up and bought my two certificates the restaurant.com site started sending emails with other offers every day or two.