There are a few ways to approach this question, but I'm going to look at it through the lens of Supreme Court decisions that explicitly discuss the question of citizenship and seek to interpret the meaning of the 14th Amendment's Citizenship Clause.

The immediate purpose of the 14th Amendment's Citizenship Clause when it was adopted in 1868 was to ensure that black Americans would be considered citizens of the United States. However, various court cases over the next few decades further clarified who was and was not covered by the language of the amendment, and what, precisely, the phrase "subject to the jurisdiction thereof" meant. (Note that the 14th amendment also provides for the "equal protection of the laws" - which is a hugely consequential clause as well - but for the purpose of this question, we'll focus on the Citizenship Clause of the 14th Amendment.)

In 1857 - a few years before the start of the U.S. Civil War - the Supreme Court had ruled in the case of Dred Scott v. Sanford (often referred to simply as "the Dred Scott decision") that black Americans were "not included, and were not intended to be included, under the word 'citizens' in the Constitution." [1] This was a highly controversial decision at the time and contributed to the sectional tensions that eventually led to the U.S. Civil War.

After the Civil War ended and the adoption of the 13th Amendment abolished slavery, the United States had to clarify whether the roughly 4 million newly emancipated black Americans were citizens, since the existing Supreme Court precedent was that they were not. The 14th Amendment was intended - in part - to settle this question. A person was a citizen of the United States by the simple fact of their birth within the United States, which meant that black Americans were undoubtedly citizens.

Numerous cases related to the 14th Amendment reached the Supreme Court over the next few decades, all seeking to clarify one or more aspects of the amendment. For the purposes of the Citizenship Clause, the two most important cases were Elk v. Wilkins (1884) and United States v. Wong Kim Ark (1898).

In Elk v. Wilkins (1884), the Supreme Court ruled that Native Americans were not citizens by right of birth under the language of the 14th Amendment because they were subject to their tribal government and not subject to the jurisdiction of the United States at the time of their birth. The decision states:

"This section [the Citizenship Clause] contemplates two sources of citizenship, and two sources only: birth and naturalization. The persons declared to be citizens are "all persons born or naturalized in the United States, and subject to the jurisdiction thereof." The evident meaning of these last words is not merely subject in some respect or degree to the jurisdiction of the United States, but completely subject to their political jurisdiction and owing them direct and immediate allegiance. And the words relate to the time of birth in the one case, as they do to the time of naturalization in the other. Persons not thus subject to the jurisdiction of the United States at the time of birth cannot become so afterwards except by being naturalized, either individually, as by proceedings under the naturalization acts, or collectively, as by the force of a treaty by which foreign territory is acquired." [2]

Native Americans were ultimately recognized as U.S. citizens by the Indian Citizenship Act of 1924. [3]

In United States v. Wong Kim Ark (1898), the Supreme Court ruled that a man of Chinese descent who was born in San Francisco was a citizen of the United States even though his parents were not U.S. citizens at the time of his birth. This decision states:

"The foregoing considerations and authorities irresistibly lead us to these conclusions: the Fourteenth Amendment affirms the ancient and fundamental rule of citizenship by birth within the territory, in the allegiance and under the protection of the country, including all children here born of resident aliens, with the exceptions or qualifications (as old as the rule itself) of children of foreign sovereigns or their ministers, or born on foreign public ships, or of enemies within and during a hostile occupation of part of our territory, and with the single additional exception of children of members of the Indian tribes owing direct allegiance to their several tribes. The Amendment, in clear words and in manifest intent, includes the children born, within the territory of the United States, of all other persons, of whatever race or color, domiciled within the United States. Every citizen or subject of another country, while domiciled here, is within the allegiance and the protection, and consequently subject to the jurisdiction, of the United States." [4]

While the original purpose of the amendment was to ensure citizenship for black Americans, the language of the amendment is clearly broader than that - as affirmed by the U.S. Supreme Court. While there are certainly those who wish to argue the meaning of "subject to the jurisdiction of" for political purposes today, the basic concept of birthright citizenship - and the meaning of this phrase - has been well established for over 100 years.

​

[1] Dred Scott v. Sanford (1857): https://supreme.justia.com/cases/federal/us/60/393/

[2] Elk v. Wilkins (1884): https://supreme.justia.com/cases/federal/us/112/94/

[3] National Archives, "Indian Citizenship Act of 1924": https://www.archives.gov/global-pages/larger-image.html?i=/historical-docs/doc-content/images/indian-citizenship-act-1924-l.jpg&c=/historical-docs/doc-content/images/indian-citizenship-act-1924.caption.html

[4] United States v. Wong Kim Ark (1898): https://supreme.justia.com/cases/federal/us/169/649/

I answered your question in a more expansive overview of the topic a little over a year ago. To copy and paste a section of that answer that's particularly relevant:

We have no written record of any debate regarding who is actually encompassed by the phrase "not subject to any foreign power." What we do have is a discussion on who is encompassed by the phrase "subject to the jurisdiction,” courtesy of the Congressional Record:

Mr. Howard: "This amendment which I have offered is simply declaratory of what I regard as the law of the land already, that every person born within the limits of the United States, and subject to their jurisdiction, is by virtue of natural law and national law a citizen of the United States. This will not, of course, include persons born in the United States who are foreigners, aliens, who belong to the families of ambassadors or foreign ministers accredited to the Government of the United States, but will include very other class of persons.”

.........Mr. Doolittle: “I presume the honorable Senator from Michigan does not intend by this amendment to include the Indians. I move, therefore, to amend the amendment—I presume he will have no objection to it—by inserting after the word “thereof” the words “excluding Indians not taxed.”

Mr. Howard: “I hope that amendment to the amendment will not be adopted. Indians born within the limits of the United States and who maintain their tribal relations, are not, in the sense of the amendment, born subject to the jurisdiction of the United States. They are regarded, and always have been in our legislation and jurisprudence, as being quasi foreign nations.”

Mr. Cowan: "...I would be glad if the honorable Senator in good earnest would favor us with some such definition. Is the child of the Chinese immigrant in California a citizen? Is the child of a Gypsy born in Pennsylvania a citizen? If so, what rights have they?......[x]

Ultimately, the answer was decided that children born in the United States to parents who are not U.S. citizens are, in fact, citizens. Via Wikipedia (used only because it's the most concise explanation I've found of the debate):

...concerning the children born in the United States to parents who are not U.S. citizens (and not foreign diplomats), three senators, including Trumbull as well as President Andrew Johnson, asserted that both the Civil Rights Act and the Citizenship Clause would confer citizenship on them at birth, and no senator offered a contrary opinion.

Now, in the context of whether 14A legally applies to a) children of immigrants and b) the children of immigrants who are in the country illegally, we have three Supreme Court cases to consider: US v. Wong Kim Ark (1898), Regan v. King (1942), and Plyler v. Doe (1982). The key court case around the issue of birthright citizenship is US v. Wong Kim Ark (1898) [full text of the opinion here].

The court ruled that a person who:

• is born in the United States,
• of parents who, at the time of his birth, aren't subjects of a foreign power,
• whose parents have a permanent domicile and residence in the United States, and
• whose parents are there carrying on business and are not employed in any diplomatic or official capacity of the foreign power to which they are subject,

becomes, at the time of his birth, a citizen of the United States by virtue of the first clause of the 14th Amendment of the Constitution.

Additionally, the court held that:

The real object of the Fourteenth Amendment of the Constitution, in qualifying the words, "All persons born in the United States" by the addition "and subject to the jurisdiction thereof," would appear to have been to exclude, by the fewest and fittest words (besides children of members of the Indian tribes, standing in a peculiar relation to the National Government, unknown to the common law), the two classes of cases -- children born of alien enemies in hostile occupation and children of diplomatic representatives of a foreign State -- both of which, as has already been shown, by the law of England and by our own law from the time of the first settlement of the English colonies in America, had been recognized exceptions to the fundamental rule of citizenship by birth within the country. The principles upon which each of those exceptions rests were long ago distinctly stated by this court.

So the Supreme Court, after a painstaking review of the law and history of the matter, explained that they concluded the phrase "subject to the jurisdiction thereof" has only three exceptions:

1. Members of certain Indian tribes;
2. Children of alien enemies in times of hostile occupation; and
3. Children of diplomatic representatives of a foreign state.

The holding of the case genuinely is that straightforward. If you are born in the U.S. and fall outside of these three exceptions, you are a citizen, and laws have since been passed that include all Native Americans as American citizens (the Indian Citizenship Act of 1924).