I don't know where the idea came from that demanding certain ID gave Republicans an edge in close elections. There is certainly an elections/campaign industry that researches this sort of information-- for the benefit of paying clients, mind you. There are also partisan think tanks that come up with rationales that cover up the partisan reason for changing voting rules. So, the GOP finds that they win elections by requiring specific IDs, and they enact these requirements on the false pretense of MASSIVE VOTER FRAUD.
It's nice to have a good post to refute this claim.
Extra. A much better conservative rundown of the evidence. Outright lies are debunked, stronger evidence presented, and innuendo fills out the roster.