People visit Ground Zero because many people lost their lives there and they wish to pay their respects to them. What happened there was horrible and people want to remember those who lost their lives needlessly.
I would say that each individual has their own reason for visiting Ground Zero. As someone above has said I would visit as a tourist just to try and reconcile in my own head the reality of what happened on that day. Also it is a way to show respect for those that were lost.
While I haven't been there, I'll tell you one of the reasons that I'd like to visit... For me, 9/11 was very unreal but very scary. I live quite a distance away and it was like watching a movie on t.v. For me, seeing the site would help me to assimilate the surreal events that unfolded on t.v. with reality.
I have visited the site on my last two trips to New York City. It is just as much of a memorial as is the Lincoln and Washington sites. In one sense, since Ground Zero is still changing, it is a curious lure to those who wish to view--and remember--the differences from before 9/11 and after.
I'll second brettd's second answer. An example: CNN's coverage of the 1991 Gulf War made the network and its reporters famous. The American public was inundated with war news, and we watched almost constantly. I remember being riveted, in spite of myself.
Of course, we can at least partially thank the popularity of that coverage for the 24-hours of spin and sensationalism CNN broadcasts today.
Two reasons, I think. One is as pohnpei argues above, to pay their respects. That is the idea that it is hallowed ground, and to visit such a site pays homage to the people who died there, and clarifies the ideals they would like to defend.
Secondly, I think Americans (and humans in general) have a fascination with, and are drawn to, sites of disaster or carnage. It's the same reason we glue ourselves to the TV set when a natural or man made disaster happens. We're horrified, but we want to watch.
I would imagine that it is for the same reason that people visit old battlefields or (closest comparison) Pearl Harbor. I think that people visit Ground Zero to pay their respects to the people who died there.
I think it is a very common nature of people and particularly when they are acting in role of tourists to be interested in seeing places which are connected with historical event. It is this instinct of people that is responsible for tourists vising Ground Zero. Perhaps people living today do not realize it, but the blowing up of the world trade center buildings an event of major historical significance, and ground zero is the place very much associated with it.