I'm aware of confirmation bias, so I'm going to reserve judgment on the study itself, but I will say this just makes sense. According to a recent study, Voter ID laws, by making vote fraud more difficult, actually protect the poor, minorities, and the elderly, rather than disenfranchising them.
And that makes sense. If, as some claim, it's so hard for people to get a photo Id, then it stands to reason that those same people are more at risk of crime in general. When you have a hard time interacting with the government, you'll have a hard time calling for help, or filing a police report, or even knowing if you've been a victim in some cases (for instance: identity theft). The same would apply to vote fraud.
What this really shows, though, is that Democrats are not concerned with people being disenfranchised- if they were, they would support some form of voter Id. It's not that they say Republicans' plans are flawed, rather they claim the laws are, by their very nature, discriminatory. That is, they believe having to prove your identity to vote is something beyond the means of a significant portion of the population.
Now, I don't believe they really believe that, which brings us back to the study. If it is correct, their motives are actually far more sinister. Rather than caring about people being disenfranchised, they oppose the laws specifically because it will suppress the disenfranchisement that is the result of the voter fraud many of them are perpetrating.