RE: Help me understand why anyone would be opposed to proving citizenship in order to vote

Opposition mainly hinges on a few things, as @preparedwombat mentions, But just to go more in-depth:

the idea that there has never been any proof of widespread voter fraud—that despite the constant complaints of it from the GOP, they have never been willing or able to show any real evidence. Not only that, but there have been numerous bipartisan investigations and court rulings have confirmed the rarity of voting fraud—so this is all a complete waste of time and resources. It is simply the Republicans pandering to their base by stoking outrage over a largely nonexistent issue.
The history of laws framed as "keeping out the illegal vote" but actually being used for voter suppression in the past and the fear that strict ID laws could be used the same way. As I'm sure you know, things like poll taxes, literacy tests, residency requirements, even seemingly crazy things like math tests have been used for this in the past, especially to prevent Blacks from voting. On the surface, simply requiring an ID seems neutral, but we have learned from history to be doubtful of such things and to expect their misuse. That may be a cynical attitude, but it is what is fueling the Democratic fears, anyway.
It seems to be aimed at people who tend to vote Democrat. Namely: low-income individuals, minorities, and college students. These people are more likely not to have an ID and face obstacles in getting them. The process of getting an ID often requires involves multiple steps—traveling to a DMV, providing documents like birth certificates, etc.—which can be particularly difficult for people in rural areas, disabled individuals, those without flexible work hours, or college students who might be in school many states away from where they need to travel to obtain the ID. None of these barriers are insurmountable, but they do put obstacles, sometimes difficult ones. Obstacles that Democrats feel are not only unnecessary but encourage people to not bother and just skip voting.
Even if IDs were easy to get, which in many cases they aren't, there is a lot of Bureaucratic burden to all this, which is made even more difficult by the fact that Republicans are constantly trying to cut funding for the very things that they want to require. For example, Republicans are constantly trying to cut DMV hours and locations in areas with high minority populations.