They have tried as hard as they could to find evidence of this widespread voter fraud, but it just doesn't exist. In the end, they just gave up and continued to tell the lie anyway (evidently subscribing to the theory that a lie repeated often enough will be believed). The following facts, from Mother Jones magazine, illustrates this point:
649 million votes cast in general elections
Here in Texas, the Republicans the Republicans thought they had found some fraud after the last election (less than 10 out of millions of votes), but all charges had to be dropped when it was discovered no fraud was involved. We had a similar case here in the Panhandle, where a woman was arrested for voter fraud. But it turned out she had recently moved and was voting in the right precinct after all, so the charges were dropped.
The Republicans have come up with a solution for a problem that doesn't exist -- but they know that. They were never really out to prevent voter fraud anyway. They were out to suppress the votes of those least likely to vote Republican. Note the chart above. All of the groups least likely to have some form of photo ID are the groups most likely to vote Democratic.
A recent state legislator in Pennsylvania bragged that the voter ID laws would allow Republicans to win the coming election. I don't think he meant for his statement to get out to the general public, but he might actually be right about his assessment. It turns out that the voter suppression law in that state could deny the vote to as many as 10% of registered voters (or about 750,000 voters). If it turns out that the election is close in Pennsylvania, losing 750,000 votes could be disastrous for Democrats.
And the situation is the same in many other states. Florida is a prime example. It is generally considered to be a toss-up state, but the Republican governor is trying to give his party an edge by purging many voters from the voter registration rolls. Here in Texas, the Republicans will probably win anyway, but the Republicans are afraid of the very fast growing minority population so their Voter ID law is a hedge against them losing power in the future.
Voter ID is not protecting American elections from fraud (which basically doesn't exist). It is a naked attempt to hold on to power at all costs -- even if they have to cheat to do it. The Republicans preach a lot about "values", but those values don't seem to include ethics.