Theory of Complexity – Why Cook-Levin Thorem's Evidence May Mean NP-Hardness

I'm studying Cook-Levin's theorem, but I've encountered a problem. The Cook-Levin theorem shows that any NPTM can be encoded as a Boolean formula. About the given language $ A $example $ w $and NPTM $ M $ Who's deciding $ A $, I understand that when a Boolean formula $ φ is true when $ M $ accepted $ w $ and wrong when $ M $ rejects $ w $, decision problem $ w∈A? $ will be the same problem as $ φ = true? $. But I am confused as to the fact that we can certify the existence of a reduction of any other problem from NP to SAT, with Cook-Levin's theorem. I can not make a real reduction from Karp to SAT without specifying the origin of the reduction, so I can not say that the SAT is in NP-Hard. Please, help me understand why conversion to Boolean formula can mean NP-Hardness of SAT …

Sorry for my bad English.