The psychology of this goes a little further for most humans (most Americans) than simply Freedom of Speech. If someone hates us so much as well as hates Christians and Jews, why would he want to build in this particular location? Or, in this Nation at all? Because of the freedoms in which we believe and revere.....to have discussions like this? The group that opposes this mosque is questioning the motives in this fashion. And, there is an opposing group that is for this mosque being built in this location because they believe it be healing to many people and on many levels.
Are you asking why would a Muslim want to build a mosque in NYC? It probably has partly to do with the larger than average number of Muslims who live in the area. The fact that people from one religion hate another is as American as apple pie. Even within the Christian religion, people within the various denominations regularly bash others and say very hateful things. I've heard Baptists say terrible things about Catholics for instance.
I have no doubt that the Muslims are trying to grow and expand their turf, but then so is everybody else. That's a motive, not so ulterior, shared by many. I also have no doubt that some Muslims would like to see this country turned into a Muslim nation. Again, the same could be said for some Christians wanting their beliefs to be the basis of government. Their reasoning I guess is that they were here first.
This is why I'm firmly in favor of a strong
separation of church and state. We all have the right to observe or disregard the sanctity of religious doctrine as we see fit, we have the obligation not to interfere with others in this regard, but we don't have to like each other. When I say "we" I'm talking about everybody, Muslim, Jew, Christian, atheist, humanist, Mormon, Wiccan, Unitarian, Satanist, Rastafrian, Pastafarian.