It means as long as you give thanks for the Sabbath you are in the new Covenant and your sins will be forgiven as you are merited a place in the world to come.
Remember when you say ata chonantanu or bein kodesh lekol Sabbath is over and we are in the Motzei Sabbath Fast. And when one finishes saying Havdalah over the Kos she shel bracha and drinks, the Motzei Sabbath fast is over!
So why instead of rushing to eat other food you should eat the shirayim and degs of Sabbath instead?
