Birtherism was around long before Donald Trump stepped into the fray. According to an article in Politico, it was actually Hillary Clinton's camp who brought the Obama origin issue out of the lunatic fringe and into the national conversation. For the next several years a chorus of prominent Republican politicians and talking heads kept the theory alive. In 2011, when he was weighing a presidential run, Trump join the ranks of the birthers well after the rumors had been discredited and many who had initially fanned the anti-Obama flames had distanced themselves from the idea.
That didn't stop Trump from going on Good Morning America, The View, and anywhere else that would have him and claiming that there were unanswered questions regarding Obama's birth. After they had had enough of Trump's badgering, the White House released the longform birth certificate. Trump responded by holding a press conference in which he proclaimed, "I am really honored, and I am really proud that I was able to do something that nobody else could do."
You might wonder why Trump went through all of this despite the mountain of evidence to the contrary. If you're curious about Trump's motives you aren't giving him enough credit. This is political theater, no different than his work in the boxing ring or alongside Stone Cold Steve Austin.
And in case you're wondering, Trump is still a birther.