To those of us who have government-issued voter IDs, asking us to show it in order to vote seems reasonable. After all, we don't want people impersonating someone else in order to cast a vote illegally.
Unfortunately, millions of American citizens -- mainly the poor and elderly -- don't need a government ID and consequently don't have one that's current. They're not all that easy to get -- you need transportation, supporting documentation, and sometimes money to pay for it -- so realistically many people won't go to the trouble just to be able to vote.
And even if they're willing to go to the trouble, they might not succeed. For just one example, here's a recent case from Tennessee: A 96-year-old African American widow who has been voting almost all her life has recently been denied a photo ID for a lack of a marriage certificate.
Furthermore, evidence of any significant degree of in-person voter fraud is close to nonexistent. For one thing, it's a very expensive and inefficient way to influence an election, and there are, unfortunately, far easier alternatives.
Some legislators may vote for these laws out of a sincere but misplace concern about electoral integrity, but it's pretty obvious that others have more practical concerns: People without IDs tend to be elderly or poor, and elderly or poor people tend to vote Democratic. As with the "literacy" tests of the Jim Crow era, a reasonable-sounding purpose conceals the actual intent and real-world result.