Individuals most likely not to have the required ID are poor, and while the ID itself may be free of charge, the burden of obtaining it is not. Once you factor in time waiting, traveling and lost wages (many people don't have paid leave in the U.S.), that ID may very well cost them $75 - $400. For those living pay check to pay check, that may be a considerable amount of money. That's not necessarily the same burden as taking time off to vote, as most individuals won't need to, as their polls open early or after working hours unlike a DMV. Most states also have time off to vote laws, and some even require that it's paid time off.
Let's say you want to obtain a voter ID in South Carolina, for example, because you have no other photo ID. You'll need to present a birth certificate. What, can't find yours? Good luck, you'll need a photo ID to obtain a new copy of that birth certificate.
Perhaps you are an 18-year old high schooler in Alabama eligible to vote for the first time. You don't have a driver's license, because you can't afford a car and don't drive. Your closest DMV is 40 miles away and only open one day out of the week. There's no public transportation, so you'll have to coordinate a ride with someone who can take you on a day the DMV is open. Previously, this voter would be able to show up to their polling site on the day of the election with a bank statement and vote.
It would be one thing if voter fraud was prevalent and all these new measures needed enacted, but there isn't. The rate of fraud in U.S. elections is close to zero. Instead, they're yet another set of hurdles voters, particularly minorities and the poor, face when asserting their right to vote.