The meaning of "subject to the jurisdiction thereof " is not very ambiguous, but one would need to delve deeper into history than manifest destiny.
To trace the origins of jus sanguinis vis-a-vis jus soli, the starting point is King Edward's de Natis Ultra Mare of 1351, and then work through the evolvements and iterations of subject to the jurisdiction up to Emerich de Vattel's The Law of Nations in 1758 and its impact on American citizenship doctrine in the founding era and the Constitution. An example is Jefferson's Virginia's Citizenship Statute of 1779:
Be it enacted by the General Assembly, that all white persons born within the territory of this commonwealth and all who have resided therein two years next before the passing of this act, and all who shall hereafter migrate into the same; and shall before any court of record give satisfactory proof by their own oath or affirmation, that they intend to reside therein, and moreover shall give assurance of fidelity to the commonwealth; and all infants wheresoever born, whose father, if living, or otherwise, whose mother was, a citizen at the time of their birth, or who migrate hither, their father, if living, or otherwise their mother becoming a citizen, or who migrate hither without father or mother, shall be deemed citizens of this commonwealth, until they relinquish that character in manner as herein after expressed: And all others not being citizens of any the United States of America, shall be deemed aliens.
Another example is John Jay's letter to George Washington dated July 25, 1787, which was the foundation for the two citizenship clauses in the Constitution.
This culminated in the debates of the 39th Congress regarding "subject to the jurisdiction," which aligned with the doctrine of jus sanguinis.