Exactly. Given to them by the Constitution.
Article I, Section. 8. The Congress shall have Power ...To establish an uniform Rule of Naturalization....To make all Laws which shall be necessary and proper for carrying into Execution the foregoing Powers,
What is the relevance of that power in the context of who is qualified to be president? I'm of a mind that the words "natural born citizen" define a class of persons that is distinguishable from "naturalized citizens".
Unless you modify the phrase "rules of naturalization" so as to avoid that distinction, and perhaps that's what you have in mind.
As it stands, anchor babies (for example, born in the US, raised in Korea or Japan or China, or Russia) are natural born citizens by dint of being born on US soil. They can be raised as Koreans (just to pick a nation) and be eligible for the US presidency. But, Congress claims it can reverse that, I believe.
Anyway, it just feels weird that Congress can effectively amend the constitution with legislation.