(Although I still don't understand how getting an ID could be so difficult in USA.)
To get a picture ID (Driver's License or State ID card, most businesses don't accept passports even though they're tougher to get - meaning there's no point in having it unless you actually plan to travel), you need to go to the nearest Secretary of State Office or Department of Motor Vehicles. If you live in a major metropolitan area, there are usually a few different locations but no matter where you are convenience of location involves some luck. Even in big cities, getting to one without a driver's license can be a bitch and half - in rural areas, even more so. So you need either public transit or a ride. Next, you have to go when they're open. The Michigan (where I live) Secretary of State office is open from 9AM to 5 PM (7PM on Wednesdays where available) Monday through Friday, though some Super! Centers are open Saturday 9AM-Noon. Once your there, you have to wait your turn. I waited 2 hours to be seen (for the 10 minutes needed to renew my Driver's License) at on a fairly average Thursday. You pretty much have to have at least half a day off of work to make going worth anything. This doesn't even get into the items required to prove your ID to get a picture ID (including a particular type of Birth Certificate copy, your social security card -which much be obtained from a different government office, some bills in your name at your address, et al). Plus the ID itself costs money, though renewing my DL late only cost about $25...though that can be a lot of money to some people.
For the working poor who don't have a driver's license (because if they had one, there wouldn't be an issue), the whole process (Taking a day off of work, finding transportation, making sure you have all the exact right documents, etc) can be a nigh-insurmountable hassle. Especially when the alternative is to simply not vote.