Short answer: Because it's important to hear the other side of the story.

However, I think it is important to elaborate on the reason why (also, the combination of overcast skies this morning, working on EC2 until late last night, and the fact that there's not much work to be done at the moment has induced a certain degree of drowsiness and desire to delve into such topics as this, rather than more C++).

Someone makes an assertion X. The question here is, how should we respond when someone claims ~X. I think the answer lies in how, if any, ~X affects our understanding of X.

If, X is an axiomatic claim, then hearing ~X may be interesting intellectually but would have no bearing on X. A well-known example, of course, is Euclid's fifth axiom. If someone declares that this is not correct, it has no effect whatsoever on Euclidean geometry. I view religious claims, such as "God exists," "God is all-benevolent", to also belong to this category.

Now, if X is a scientific statement (or theory), e.g. "no body with non-zero mass can travel faster than light in vacuum", the principle of bivalence applies (simplistically speaking), so X is either true or not. In this case, hearing ~X is not only important, but vital.

These two were the easy cases. Analysis of statements in other disciplines ("soft sciences", to use a slightly derogatory term) such as history, sociology, etc. is muddied by at least two factors: (i) There are many more factors that are hard to isolate and test in any setting, (ii) the concept of "truth" has been challenged in liberal arts as a condescending construct of dominant culture. In these cases hearing the argument for ~X may lead to a synthesis with X, whereupon you arrive at a "better", joint understanding, X'. So, again hearing ~X is important, but leads to a slightly different outcome than the scientific case (those who deny that each newer scientific theory gets closer to "truth", e.g. Kuhn, would not agree that there's a difference).

My totally subjective view is that, although all three are historic statements, "Americans landed on the moon" should be analyzed by the scientific approach Moon landing is a well-defined event and there are few factors to take into consideration that would make its detection difficult.

"US government was behind 9/11" is on the other end of the spectrum. The assertion is unbelievable (in fact, I find it ridiculous) but the many complexities of the situation and definition of terms (e.g. do you think US's activities in the Middle East precipitated the events, does that count as "was behind"?) would make one want to engage with proponents of these views, just to see if any important facet of the whole set of complex events is missed. I might learn and advance my understanding of the event from such discussions. (Sadly, this almost never happens, generally such people are politically motivated ideologues who parrot useless propagandist nuggets of falsehoods).

The interesting question is: where to place the Holocaust denial in this spectrum? Personally, I put it much closer to the moon landing argument than the 9/11 one.

This, however, by no means be the basis of making it illegal to voice such opinions, as it is in some European countries or censure them. My motto is simple: Everything should be discussable and analyzable, there should be no intellectual taboos whatsoever in a modern society.