So this is a great question with a really interesting answer! In short, it was U.S. Grant that came up with this tactic during the Vicksburg campaign, and humorously enough, Sherman thought Grant was crazy for suggesting it! (He later came around, as you know = )

So, let's set the scene. It's late-1862, and Grant is back in the good graces of the Union Army, and as a result is given the very important and difficult task of capturing the Confederate city/river fort of Vicksburg. The Union forces are on the opposite (west) side of the Mississippi River from Vicksburg, and can't gain a foothold on the eastern bank to attack the city. Nothing seemed to work, and until Vicksburg fell, the Union couldn't gain complete control over the Mississippi River. Jefferson Davis as well as Lincoln understood that the loss of this important waterway would effectively split the Confederacy in two, so Vicksburg really was the key to landing a crushing Union blow. Grant tried seven different approaches to take Vicksburg over the course of about 6 months, which included everything from a frontal assault, an expedition down a nearby river (the Yazoo) to flank it, and even re-digging the Mississippi River so that they could just sail around the city (this did not work).

Grant's solution proved what an insightful, creative military mind he possessed, and should dispel any Lost Cause myths about the man being a numbers-reliant butcher. Grant's solution was to march his army down the west bank of the river, through the swamps and beyond the reach of the city's guns, cross the Mississippi, take the city of Jackson (which was on Vicksburg's eastern side of the river and the city's main supply point via a rail line), and encircle Vicksburg.

This isn't as easy as it may sound, though! To go this route, Grant couldn't bring his wagons or reliably extend his supply lines, as it was too swampy, and he wouldn't have been able to get anything substantial over the river when they crossed (at least in the volumes needed to sustain him). This was all aside from the fact that Grant's forces had to move fast enough to steal this march on the Confederates, because if they caught wind of what he was doing and he hadn't executed his plans fast enough, reinforcements from Gen. Joe Johnston's army could have swooped in to annihilate Grant's split forces. The whole plan relied on Grant being able to move fast with just his troops, not his wagons, so he had to figure out how to allow troops to fight for weeks with just a couple of days worth of rations (whatever they could carry).

This is where the strategy took shape, though: the strategy that would make Sherman famous. Grant figured that his army could live off the land when they got to the eastern side of the river, and whatever they didn't carry could be foraged from the locals. This turned out to be a very accurate prediction, and although Sherman originally advised against Grant's plan, he was the first to admit he was dead-wrong when Grant pulled it off. Indeed, Sherman was eager to put the small-scale experiment to the test when Grant put him in charge of the Union's western armies, and once again, the strategy proved effective. When Sherman's tactics also had the bonus effect of lowering civilian morale and robbing the Confederates of much needed resources, the modern total war strategy as we understand it today was born.

[Sources: Bruce Catton, 'Grant Takes Command' & 'Grant Moves South'; Doris Kearns Goodwin, 'Team of Rivals'; James McPherson, 'Battle Cry of Freedom']