In the U.S., natural born citizenship originally came from U.S. "common law", which is based on British common law. The "common law" is sometimes misunderstood to be some kind of "implied" law because "that's the way it's always been done", but that's not exactly correct. The "common law" is the set of court decisions and judicial precedents that have previously been decided that aren't subsequently passed by acts of the legislatures and written into the law books themselves, but are nevertheless followed as laws as if they are. As a recent example in U.S. history, the Supreme Court decision Obergefell v. Hodges forbid discrimination of issuing marriage licenses to same-sex couples, so while Congress never passed a federal law making same-sex marriage legal, it is nevertheless legal under common law because of the Supreme Court ruling. A hundred years from now, it will still be the law even if Congress never passes an explicit act legalizing same-sex marriage in the U.S. federal code of laws. It's a common law, as long as it's not reversed by a later court ruling. And this doesn't only apply to Supreme Court rulings--any ruling by any court that has the effect of a precedent is considered a common law.

When the U.S. was founded, instead of writing out all these British common laws into the U.S. law books, Congress relied on Great Britain's common law as the basis in U.S. law. This allowed them to sidestep having to re-write all kinds of minor things that they didn't think needed further mention or clarification, and moreover, it prevented them from having to re-litigate hundreds of years of judicial precedent.

To that end, the Founders relied heavily on the multi-volume Commentaries on the Laws of England written by Sir William Blackstone, which collected the most important common laws into a single work. You will see the book referenced from time to time in the writings of the Founders: among other places, John Adams wrote about it in his diary, Alexander Hamilton mentioned it in Federalist Paper No. 84, and James Madison listed it among the books for use by Congress when the Revolutionary War was ending and Congress was going to soon be debating a permanent U.S. Constitution.

Thomas Jefferson wrote a brief, incomplete autobiography of himself and actually mentioned why the Founders used Blackstone's common law book rather than starting anew. He recounts that in 1777, a committee of a few members of the Continental Congress discussed the issue:

"The first question was whether we should propose to abolish the whole existing system of laws, and prepare a new and complete Institute, or preserve the general system, and only modify it to the present state of things...[the legislature] had been in the practice of revising from time to time the laws of the colony, omitting the expired, the repealed and the obsolete, amending only those retained, and probably meant we should now do the same, only including the British statutes as well as our own: that to compose a new Institute like those of Justinian and Bracton, or that of Blackstone, which was the model proposed by [Congressman] Mr. Pendleton, would be an arduous undertaking, of vast research, of great consideration & judgment; and when reduced to a text, every word of that text, from the imperfection of human language, and it's incompetence to express distinctly every shade of idea, would become a subject of question & chicanery until settled by repeated adjudications; that this would involve us for ages in litigation, and render property uncertain until, like the statutes of old, every word had been tried, and settled by numerous decisions, and by new volumes of reports & commentaries; and that no one of us probably would undertake such a work, which, to be systematical, must be the work of one hand."

In other words, the Founders did consider re-writing the whole of common law, but they realized the impracticality of it, because it would open up every law to a new court challenge, when the law had already gone through that process, and been settled, which is why it was considered the common law. There was no need to go through all that headache all over again, and in any case, nobody wanted to try to take on the mammoth task of writing out such a book of common laws, when it was simpler to rely on the volumes that had already been produced.

So, they relied on Blackstone's multi-volume book. And this is what Blackstone's book says about natural born citizenship:

"§ 501. a. Natural, or perpetual allegiance. — Allegiance, both express and implied, is, however, distinguished by the law into two sorts or species, the one natural, the other local; the former being also perpetual, the latter temporary. Natural allegiance is such as is due from all men born within the king's dominions immediately upon their birth. For, immediately upon their birth, they are under the king's protection; at a time too, when (during their infancy) they are incapable of protecting themselves. Natural allegiance is therefore a debt of gratitude ; which cannot be forfeited, canceled, or altered, by any change of time, place or circumstance, nor by anything but the united concurrence of the legislature. An Englishman who removes to France, or to China, owes the same allegiance to the king of England there as at home, and twenty years hence as well as now...."

Further:

"§ 503. c. Allegiance is personal. — ...This allegiance, then, both express and implied, is the duty of all the king's subjects, under the distinctions here laid down, of local and temporary, or universal and perpetual. Their rights are also distinguishable by the same criterions of time and locality; natural-born subjects having a great variety of rights, which they acquire by being born within the king's ligeance, and can never forfeit by any distance of place or time, but only by their own misbehavior: the explanation of which rights is the principal subject of the two first books of these Commentaries. The same is also in some degree the case of aliens; though their rights are much more circumscribed, being acquired only by residence here, and lost whenever they remove..."

There's much more detail in those pages of Blackstone's Commentaries distinguishing natural rights vs. alien rights if you care to get more context.

But the main point is, natural-born citizenship had been established under English common law, which was then adopted as U.S. constitutional law, if left unwritten so that it didn't have to be re-litigated, just like the whole of the rest of common law. But surprise surprise! A court challenge to determine whether these rights extended to free black people under the Constitution reached the Supreme Court and they ruled in Dred Scott that, no, free black people did not have natural-born rights under the federal constitution. That's because there's stuff in the Constitution that considers some black people property, so the court ruled:

"A free negro of the African race, whose ancestors were brought to this country and sold as slaves, is not a 'citizen' within the meaning of the Constitution of the United States."

As has been regularly argued since, that's definitely a leap in logic, and a consequence of leaving the common law of natural rights implied from previous English judicial precedent rather than writing it out explicitly in the U.S. Constitution or federal code.

(Side note: Dred Scott did not prevent free black people from being considered citizens of free states, but this didn't help in interstate matters, such as when a slave state claimed that some black person was an "escaped slave" and must be returned to bondage to the slave state. Since the federal government didn't consider the free black citizen in the North natural-born with natural rights, they could not file a lawsuit in federal court to stop the slave state from taking them South. This is why Dred Scott was such a big deal. It trampled all over the free states' ability to protect their own citizens, and called into question just exactly how "free" black people really were even if they lived in one of the so-called "free" states in the U.S.)

In the end, the 14th Amendment needed to be passed in order to overrule the Supreme Court's decision that the common law of natural-born citizenship did not apply to people descended from African slaves. The common law applied to everyone born in the United States, black people and former slaves included.

The Origins of the American Constitution: A Documentary History by Michael G. Kammen

The Dred Scott Case: Its Significance in American Law and Politics by Don Edward Fehrenbacher

Origins of the Dred Scott Case: Jacksonian Jurisprudence and the Supreme Court, 1837-1857 by Austin Allen