First, let's get this out of the way. If you take seriously, even for a minute, the thoroughly risible conceit that Barack Obama is not an American citizen or has forged credentials to that effect or some nitwit variant of the above, then I'll also wager that no amount of evidence to the contrary will persuade you otherwise. Because you'll always find a way to dismiss it.
I spent a couple of years in the late Nineties as a regular reader of the alt.revisionism newsgroup, wherein Holocaust deniers posted one tired variation after another on the usual demonstrably untrue conceits. This is not the place to go into the details, but the short version is that the only thing that really changed on their end was the names. The folks on the other side — the ones posting rebuttals based on historical fact, the folks who took the time to demolish the arguments point-for-point — did not believe for a minute they were going to break the faith of the converted. They were doing this so that other people who had wandered in and were perhaps on the fence would not get suckered by fallacious arguments and a version of history with most of its pieces missing or reinstalled upside down.
I see something of the same dynamic here. Those who believe will always find a way to believe, and will typically do this by falling back on support from a like-minded group. The nature of the belief isn't important — it could be the old canard that we never landed on the moon (which got a major bump in attention recently, much to my disgust), or a flat/hollow earth, or anything, really. What matters most is that you have a need to believe, and that without the expression of that need you feel as if some major component of your self has been denied. (I'm willing to bet the same mechanic applies to aging rock stars who get themselves back onto the stage for one more gig, because without a stage and an audience and a band behind them, what's left? Well, if you're Brian May, a career in astronomy, but he's the exception and not the rule.)
Why do they do it? For solidarity, I think; as a way to claim membership with a corpus and not feel marginalized. People disturbed by the idea that a black man is president; people upset that anyone not of their pet political persuasion is in power; people who hate the system on general principles and look for any way they can put thorns in its side. The issue itself is never the issue; it's always a symptom of something larger, deeper, more ... conspiratorial.
The larger question of whether or not there are better things to occupy one's time, energy, attention and remaining brain cells with remains unanswered.