Because he's good.
I'm not sure at what. I listened for several minutes. I heard him say it was not up to Obama to prove that he is a natural-born citizen but rather it is up to others to prove that he is not. First he is not a natural-born citizen by definition unless he has misled everyone as to who his father is. His father was not a US citizen. End of story.
But did you ever enter a situation where US citizenship was required and it wasn't incumbent upon you to show evidence that you were a US citizen? The idea that the other guy, frequently the US Government itself, should be required to demonstrate that an applicant is NOT a citizen is one that only a Space Cadet would have. (Note that the requesting agency may attempt to VERIFY the submitted documentation and may reject it; but they will never act on no documentation at all.)