IANAL but I believe the rationale is that they need your address to ensure you vote in the proper local elections. I am a huge proponent of Voter ID especially as a voter ID law would have prevented from someone else voting under my name in one election BUT this case is not what Voter ID laws should do. They should merely be a means to ascertain that you are who you claim to be. That’s it.
Because Dems don’t care who votes: citizens, non-citizens, convicted felons, people committing voter fraud, people who can’t poke a sharp piece of metal through pre-perforated paper, etc.
In fact, the ONLY time I can think of that Dems opposed voting rights was in 2000 when they tried to exclude absentee votes from military personnel in Florida.