It doesn't in Massachussetts:
Massachusetts Marriage License Information, Laws, Requirements
What are the residency requirments for Massachusetts?
None, you do not have to be a previous or current resident of Massachusetts to obtain a marriage license that is valid statewide.
So, once again, it appears the law in Massachussetts is what their Supreme Court deems it to be, in spite of clear legislation to the contrary.
And that link you supplied shows that cousins may marry in Massachusetts, something that is not allowed in some other states.
That's what the original law addressed, it just happens to logically apply to "same-sex unions" as well. ;-)
This court decision just upheld a 1913 law banning people from marrying in Massachusetts if that marriage would be illegal in their home state.
The link you provided goes to a commercial website which may not aware about the 1913 law. Since the law has been there for long but didn't seem to be enforced for long, people may not aware or simply ignore it. The governor is quite smart. He simply refers to the law, which apparently is still in the book.