There is this ongoing mistake conservative politicians make time and time again. Their base is composed in large part of drooling idiots, and apt to believe nutty conspiracies everyone else has widely debunked and discounted. In the '60s they believed flouridated water was mind control, then in the '70s they believed black militants were rioting in the streets, then in the '90s they thought Bill Clinton was running drugs for the new world order. And now they believe probably the nuttiest thing ever: That President Obama is a secret muslim Kenyan terrorist manchurian candidate.
For Republican pols, it makes sense for them to keep these fears stoked - it brings money in and gets a reliable vote out. But the birthers want more. They believe that the presidency is completely illegitimate (not in a Supreme Court delivered the election way) and a vocal amount of them support legislation and violence to remedy what they see as a problem. Republican pols aren't backing down, with nutters like Sen. Inhofe indicating he believes that they have a point. These yahoos have already gone further than the left-right spectrum of 9/11 truthers ever went, getting people to actually sign on to legislation.
After elections its natural for parties to look at themselves in the mirror and figure out what needs fixing. The Democrats decided to be true progressives and expand their playing field to all 50 states. The Republicans have decided... to demand birth certificates.