That has an easy and uncomfortable answer: to check that all registered voters are actually citizens. And this is why Democrat run states refuse to share that database, because it might show they have non-citizens voting. I guess the same could be said about Republican run states, but those seem like they have a lower rate of illegal immigrants.

You could people just show their ID right before voting and you would not need such lists? So no illegal person could vote, right. I don't get it.

Requiring voters have identification is very controversial in the US. The Democratic party generally opposes it. Even in states requiring ID, there are almost always options to bypass it (by signing an affadavit, for example), and in almost no case does an ID prove citizenship - the US doesn't actually have a "US citizen database" anywhere, and people can be legal citizens with a right to vote with no ID.

In the US you can get a driving license without being a citizen. And that is accepted as proof of ID pretty much anywhere. That's the rub.