Well, I can give you a great number of reasons how we know what they had in mind, but it will take a long time to go through them all. How about we start with Aristotle? This is what he said 2500 years ago:
Who is the citizen, and what is the meaning of the term?...Leaving out of consideration those who have been made citizens, or who have obtained the name of citizen any other accidental manner, we may say, first, that a citizen is not a citizen because he lives in a certain place, for resident aliens and slaves share in the place;
...But the citizen whom we are seeking to define is a citizen in the strictest sense, against whom no such exception can be taken, and his special characteristic is that he shares in the administration of justice, and in offices.
...a citizen is defined to be one of whom both the parents are citizens;
How about Mathew Bacon? (From an English Law book owned by John Adams and his son John Quincy Adams.)
Note, the reference to the Parents comes BEFORE reference to place of birth.
I have been arguing this subject for several years now, and I can show you all sorts of bits and pieces that demonstrate the founders did not regard birth on the soil as the deciding factor (Indians, Slaves, Diplomats and British Loyalists after the war were all born on the soil, yet none were considered citizens.) but instead required that the Father had to be an American to make the child an American. (Till the Cable act of 1922 when women were first allowed to transfer citizenship to their offspring. Prior to this time, women were automatically the same citizenship as their husbands.)
Here's a piece from 1811 in which James Madison (writing under his pen name "Publius" says the Father must be an American to make the son American.
I can go on and on with examples, and I will if you want to see more.
Here is a Law passed by Maryland (1784, three years before the Constitutional convention.) designating the Marquis de La Fayette as a "Natural born citizen" and his male heirs forever. (Note they do not require them to be born in America. Place is not the condition that makes one "natural born." )
Chief Justice Marshall:
Vattel, who, though not very full to this point, is more explicit and more satisfactory on it than any other whose work has fallen into my hands, says"The citizens are the members of the civil society; bound to this society by certain duties, and subject to its authority, they equally participate in its advantages. The natives or indigenes are those born in the country of parents who are citizens. Society not being able to subsist and to perpetuate itself but by the children of the citizens, those children naturally follow the condition of their fathers, and succeed to all their rights."
I'll post more as you like.