They cite the elastic clause in conjunction with this line:
US Constitution said:
The actual Enumeration shall be made within three Years after the first Meeting of the Congress of the United States, and within every subsequent Term of ten Years, in such Manner as they shall by Law direct.
The question here is, what is meant by, "in such Manner as they shall by Law direct"? The above webpage essentially argues that it sanctions pretty much any question they may ask, and they briefly cited debates among the Founders/Framers about what kind of additional questions they should pose. Honestly, even if I don't like it, I find their argument hard to totally refute on this issue, especially given the historical context they provide.
Personally, I still think they may be incorrect, even if it's probably impossible to convince people of it. The explicit wording of the Constitution itself sounds to me that Congress is merely authorized to legislate the logistics of carrying out an enumeration (head count). The reason is, the Constitution explicitly mentions the information to be recorded by the enumeration - like separate counts of men and women, etc. - in the same article and section (altered by the Fourteenth Amendment). This seems to imply that the subsequent "in such Manner as they shall by Law direct" phrasing refers to the matter of logistics alone. For instance, it seems to me to refer to legislating issues like: Do we ask questions of the head of household eye-to-eye, or do a head count of each person eye-to-eye, or ask the head of household to fill out a census form, or what? The "Necessary and Proper" clause shouldn't really sanction additional questions either, since additional questions are clearly
not necessary to perform the enumeration task as directed by the Constitution.
Still, I think we should count our lucky stars that the "expansive interpretation" merely includes asking more probing questions, rather than obtaining the minimal authorized information in increasingly intrusive ways. After all, strictly speaking, taking the Constitutional wording literally could be a bit dangerous on the census issue: It seems to imply that ANY manner of enumeration is appropriate...like running people through a Soylent Green processing plant and counting the resulting volume of people snacks.

Obviously, you could make a lot of Bill of Rights arguments against that one, but still, the point is that it could probably be a lot worse.