Think Progress details how the Constitution allows for this to happen.
Under the 14th Amendment, “[a]ll 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 language is unambiguous; it grants citizenship to all persons born in the US unless they are not subject to American “jurisdiction” — a very narrow exception that applies only to children of foreign diplomats and a handful of other people. Moreover, in U.S. v Wong Kim Ark and again in Plyer v. Doe, the Supreme Court firmly rejected the notion that persons born in the US are not citizens, regardless of the immigration status of their parents.
Rand Paul is either completely ignorant of the 14th Amendment to the Constitution ( a document he allegedly loves and has read ) or he is taking the standard conservative route - overly simplistic solutions that "sound good" to your base but can't function as a real world solution.
One almost gets the feeling that Paul is testing the waters just to see how far Right Kentucky voters are willing to go. I've spoken with several staunch conservatives that I work with about Paul's views and his associations and I continually get the feeling that Republicans here either aren't willing to admit that they will vote for him or are thinking that he's so far to the Right that he's making conservatives look really bad.
Considering how Paul is fairing in Kentucky right now - he's only got a 4 point lead against Conway right now - I'm waiting for Fox"News" to do their big push in the Fall prior to the election. Are they going to stand with Rand Paul or completely ignore him? Considering their love for the Tea Baggers, they might want to tread lightly with this one.