IIRC it has to do with how easy/hard it is to get said ID. The claim is that IDs in the US are much more difficult to get for some demographics than others due to time/monetary costs and/or the difficulty of acquiring the required documents, so requiring an ID to vote would mean that otherwise eligible voters may be prevented from voting.

Some other countries have a nationally issued ID that citizens are given for little or no cost. In those cases, voter ID laws would be much less controversial. I don't know whether this is the case for all countries where voter ID laws are not controversial.

Just to be clear, I am in favor of free id cards and requiring an id card (either the free one or a driver's license) if you want to vote.

Voter ID laws would still be controversial in the US because people would say that poor people would not be able to make it to the place where they can pick up their id card. Poor people are less likely to have a vehicle and are more likely to work multiple jobs causing them to work past closing time at the DMV or wherever they would get the ID.

The other issue is that it is not actually clear if voter id laws actually cause people to not be able to vote. https://www.vox.com/identities/2017/3/15/14909764/study-vote...

It's because the US doesn't have a national ID system and a significant fraction of the population doesn't have an ID and can't get one easily.

This isn't even a secret or anything, GOP operatives openly admit that the purpose of these voter ID laws is to suppress minority voters: https://www.thedailybeast.com/republicans-admit-voter-id-law...