This is crystal-clear to me. What's the problem?
Compared to some other 18th century documents I've read, the US Constitution is downright spare with commas.
Try reading the Constitution out loud, taking appropriate breaks at the punctuation. It becomes a lot easier to understand and read; most (but not all) of the ambiguity will disappear.
The grammar rule is that you can remove a clause within the commas - so this becomes:
No Person except a natural born Citizen at the time of the Adoption of this Constitution, shall be eligible to the Office of President.
So unless you were born sometime before 1776 you aren't going to be hitting the campaign trail