How do 'register your credit card with us, and earn cashback' schemes work, or make money themselves?

For example, with T-mobile, I can register my credit card, and if I use that credit card at any of the participating outlets, I can earn cashback. Similarly, if I register a non-Bilt credit card with Bilt, then I can earn Bilt points on 'eligible spend'. How are they able to access my credit card transactions? and are they...