There is no truth to that at all, full stop. Here's the abbreviated history of birthright citizenship in the United States.
At independence, the United States inherited the citizenship laws of England. Those were established in Calvin's Case, 77 E.R. 377 (1608). Summarized, the court in Calvin's Case held that if a child is born a subject of the English king, they are entitled to the protections of the laws of England.
When the United States declared independence, the state legislatures passed reception statutes establishing that the laws of England still applied. New Jersey's 1776 Constitution, section XXII, for example, looked like this:
That the common law of England, as well as so much of the statute law, as have been heretofore practiced in this Colony, shall still remain in force, until they shall be altered by a future law of the Legislature; such parts only excepted, as are repugnant to the rights and privileges contained in this Charter;
Between 1776 and 1857, this situation is more or less stable. Then, in 1857, the Supreme Court decides the infamous case of Dred Scott v. Sandford, 60 U.S. 393 (1857). Among other things, Dred Scott presents a pretty straightforward question of procedure, because the federal courts are only entitled to hear cases between citizens of different states. If Dred Scott himself was not a citizen, he would have no right to file suit, and the case goes away. This is exactly what the court did. The Dred Scott court said that blacks were not citizens, should not be citizens, and could never be citizens, whether free or slave.
Dred Scott becomes one of the major causes of the Civil War.
After the Civil War, Congress passes the Civil Rights Act of 1866, 14 Stat. 27-30 (1866), which says 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." This is under Congress' power to declare a "uniform rule of naturalization" pursuant to Article 1, section 8 of the Constitution.
"Beyond question, by that act, national citizenship was conferred directly upon all persons in this country, of whatever race (excluding only 'Indians not taxed'), who were born within the territorial limits of the United States, and were not subject to any foreign power." United States v. Wong Kim Ark, 169 U.S. 649, 682 (1890).
As the Supreme Court also noted in Wong Kim Ark, 169 U.S. at 675, "Congress, shortly afterwards, evidently thinking it unwise, and perhaps unsafe, to leave so important a declaration of rights to depend upon an ordinary act of legislation, which might be repealed by any subsequent Congress, framed the Fourteenth Amendment of the Constitution, and, on June 16, 1866, by joint resolution, proposed it to the legislatures of the several States, and on July 28, 1868, the Secretary of State issued a proclamation showing it to have been ratified by the legislatures of the requisite number of States."
The 14th Amendment provides that "All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside." This re-establishes the rule of birthright citizenship in the United States. The 14th Amendment is explicit about birthright citizenship because Dred Scott stripped the rights of citizens from free blacks.
The clause that Trump et al. sometimes cite as a counterbalance, the "subject to the jurisdiction" part, is much narrower than most of the GOP candidates say it is. "Subject to the jurisdiction thereof" was meant to exclude two categories of people from American citizenship. Those are (a) Indians, who were treated as "independent nations" at the time, and (b) children of foreign diplomats. Wong Kim Ark, 169 U.S. at 705.
The Wong Kim Ark case is actually directly on point here, as Wong was born in San Francisco to two Chinese parents, neither of which were American citizens. Wong Kim Ark isn't an outlier, either. Previously, in In re Look Tin Sing, 21 F. 905, 910 (C.C.D. Cal. 1884), the California District Court held that birthright citizenship is something that attaches to American birth, full stop. When discussing the citizenship status of the child of a Chinese immigrant, the court said, "It is enough that he was born here, whatever was the status of his parents." For over a hundred years, the right of citizenship has attached to the fact of the child's birth here, and has nothing to do with parentage.
(on my phone, can provide sources on request, though most of what I am saying is easily verifiable)
I am not aware of any evidence that it was not intended to grant birthright citizenship, and indeed, I find it hard to believe that it could be interpreted any other way.
But to pick up on the one kernel of truth that Trump is drawing from, the explicit, stated motivation was to ensure that freed slaves and their children would be granted citizenship. This is different from the dialogue around birthright citizenship today, which centers around immigration. In that sense, you can say that the amendment was not created with today's particular dialogue in mind, simply because the political climate and most important issues were different then.
If it sounds like I'm splitting hairs, it's because I am. But Trump isn't an idiot, no matter how he tries to portray himself in the media, and I'm willing to bet anything that he's intentionally twisting the "technically true" part of this statement into something outrageous that he can use as a political tool.
Related, but perhaps not to this sub question: How important is the perceived "intent" of the authors of the constitution vs what was written? It seems like intent acts as a means of warping or filling in gaps via interpretation perhaps to conform to certain political views.