Birthright Citizenship simply refers to the principle that if you are born in the US then you are a US citizen. This has been the traditional interpretation of the 14th Amendment which states:
“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.”
Of course Trump and the other Republican nominees that are calling for an end to Birthright Citizenship don't read the Constitution,
Or, if they do read the Constitution, they don't read case law.
The argument is that the phrase "subject to the jurisdiction thereof" does not apply to the children of illegal immigrants.
This issue was resolved in 1898. In the landmark case “U.S. v. Wong Kim Ark” the Supreme Court decided that the 14th Amendment must be interpreted based upon English Common Law and included all native-born children except for those who were: (1) born to foreign rulers or diplomats, (2) born on foreign public ships, or (3) born to enemy forces engaged in hostile occupation of the country's territory.
So, unless someone would like to claim that illegal immigrants are "enemy forces engaged in hostile occupation" ending Birthright Citizenship would require either (a) the Supreme Court to overturn “U.S. v. Wong Kim Ark” or a constitutional amendment.
I suspect that none of the Republican bozos supporting this idea would bother to go this far. They're just playing off the god awful ignorance of the Republican base.