As I understand the story, he was born in Kenya and was issued a certificate of live birth in Hawaii. His mother travelled to Kenya with his father for a little while. She was so close to term that she wasn't allowed to board a plane back to the States for fear of giving birth on the plane. Supposedly his paternal grandmother had been interviewed and stated that she was there and watched him being born in Kenya.
Aside from that, there is another story about his adoption by his stepfather in Indonesia which caused him to lose his US citizenship as a child. If he had gone through the proper paperwork when he came back he would be a citizen, but a naturalized citizen and still not eligible to be President. This story claims that he is not a legal citizen to this day. There is more surrounding that story but I gave the nutshell version.
Who knows the truth. I do agree he could make it all go away by just giving the documents that were asked for and end it. If you ain't got nothin to hide, then don't.
As far as the birth certificate that was posted on that website, I saw a video where a guy broke it down and proved it was a fake. Something about how the Hawaiian birth certificates were done differently then from now and some seal. Don't remember it all, but it's pretty convincing. It's on YouTube if anyone wants to track it down.