It's synonymous with the 1866 Civil Rights Act in the first sentence.
"That all persons born in the United States and not subject to any foreign power, excluding Indians not taxed, are hereby declared to be citizens of the United States; and such citizens, of every race and color, without regard to any previous condition of slavery or involuntary servitude, ..."
Congress went ahead to pass and send to the states for ratification of the 14th Amendment because they didn't want the 1866 Civil Rights Act to be easily overturned by future legislation.
"not subject to any foreign power" = "subject to the jurisdiction thereof"
Consider too that according to Mexican law, every child born to a Mexican anywhere in the world is a Mexican national. Therefore, Mexican anchor babies are not subject to our jurisdiction, and their mothers ought to be sent down to the nearest Mexican consulate to register their babies. Or go home to Mexico.