Several things.

The most important one (in my opinion) is the problem that the companies that sell voting machines aren't making enough money and their profits would be improved if enough county election boards were convinced (or required, though state or national legislation) to buy new systems.

Another very significant problem that electronic voting is attempting to address is that election officials are evaluated (to some extent) by the public based on what appears in the newspaper on just one or two days per year -- and THAT depends largely on how quickly results are reported. So a tool that takes 2 hours to report results will look much better for election officials than a tool that takes 48 hours, even if the second produces elections that are more trustworthy and there are 2-3 months before the new candidates actually need to take office.

Another problem is that existing methods using paper look "old-fashioned". A fancy system that takes video images of the voter's face to identify them and then allows them to vote from overseas using a cell phone may not be as reliable or secure as mailing in a paper ballot, but it sure FEELS modern.

I guess my point is that there are definitely reasons these machines are being used, but "improved security" or "transparence and thus confidence in the reported results" are not the motivating forces.

The 2000 presidential election was a big driver. No one wanted to spend November of an election year hearing about "chads" ever again, so the Help America Vote Act was passed. Among other things it provided federal funding for states to replace problematic punch-card voting machines. Electronic voting, in theory, could be easier to use than a paper ballot and can automatically check that the ballot is filled out correctly (e.g. not voting for multiple candidates for a single office). And of course there were companies lining up to help states spend those federal dollars on overpriced touchscreen PCs running Windows XP.

Not enough taxpayer money spent on things that aren't broken enough.

Not saying the solution is valid or the problem is pressing, but they're trying to fix counting issues like "hanging chad".

Potentially reducing cost while increasing voter turnout

Animal spirits.