Imagine you're black living in some small town in Alabama in the 60s.

You go to vote and present your ID. The guy running the polling place takes it, looks at it, and hands it off to another "good old boy" who takes it to a back room -- presumably to destroy it. The first guy turns to you and says "You need to present your ID to vote." Indignant, you turn and see the Sheriff ... who you know is also a high ranking member of the local KKK chapter.

You have no recourse. Even if you can prove you possess a valid ID you can't prove in court that you brought it with you that day, and multiple "witnesses" will say you didn't.

That is a real thing that frequently happened. The current system takes that weapon away from the corrupt local governments and institutions. The most important thing it gives someone is time to seek recourse before voting day arrives: particularly a courts. And if the lower courts are corrupt the person being disenfranchised can appeal to a higher one.

A court can determine you're eligible and order a local government to let you vote ... and anyone that violates that order is going to have a very bad time. Sheriff Dipshit knows what he can get away with, and disobeying a federal court's order isn't one of them.