To understand the voter ID laws currently being proposed and implemented in many states, it's important to first understand the problem: Voter fraud. Voter fraud is, essentially, any effort to rig the election in order to make sure it doesn't represent an accurate total of votes from the citizens of the country in question, usually to the benefit of the fraudster.
Voter fraud takes two basic forms. The first is the casting of additional ballots for one candidate that are not legitimate, inflating the total number of votes for that candidate to give them an edge. (Known as "stuffing the ballot-box" in some circles, as the oldest and most primitive form of the practice involved literally shoving fake ballots into the boxes used for counting.)
The second is the denial or destruction of legitimate votes cast for one's opposition, in order to artificially deflate the vote totals against the fraudster and give them an edge. Sometimes this is done through literal destruction or shredding of ballots, but most often it takes the form of erecting a (literal or metaphorical) fraudulent barrier to voting. For example, a fraudster might circulate a flyer among their opponent's supporters stating an incorrect time or place for polling, hoping to fool those supporters out of casting legitimate votes.
Currently, Republicans are claiming that the best way to prevent voter fraud of the first type (illegal additional ballots cast) is to erect a (metaphorical) barrier to voting, in the form of stringent requirements for voter ID. These requirements are estimated to prevent somewhere in the neighborhood of three thousand times as many legitimate votes from being cast as they will stop illegitimate votes from going through. (Depending on which estimates are used. Some estimates suggest that the number may be as high as thirty thousand, not three thousand.) Coincidentally, these legitimate ballots generally belong to demographic groups that vote overwhelmingly Democratic.
So to make it very simple: Republicans are currently insisting it's better to commit voter fraud 30,000 times over in ways that benefit them than to stand by and allow 10 cases of voter fraud that may benefit their opponents...or them.
What's complicated is why anyone believes these crooks anymore.