Great question! Franklin is arguably one of the most important battles of the Western theater, if not the entire Civil War.

The Lead Up:

John Bell Hood was a Texas plantation owner who prized honor quite highly. He was rather aggressive in combat, which endeared him to Jefferson Davis. In turn, Davis disdained General Joseph Johnston, who had command of the Army of Tennessee outside Atlanta in 1864. Johnston began a series of confrontations with Sherman's army beginning in the spring of that year as the Union general marched on Atlanta. The two generals were well-matched - both prized maneuver warfare over direct assaults. Most of the battles during the 1864 Atlanta Campaign ended as tactical Union victories, but Johnston was playing the long game. He knew that if he could inflict major casualties on Sherman's army and slow his advance to Atlanta, Union forces might not take the city before the Presidential election in November. By denying Sherman his prize, Johnston hoped that low morale in the North might lead to Lincoln's defeat at the hands of former Union General George McClellan, who might be more favorable to the South in any peace talks. So Johnston engaged Sherman in a series of frustrating, inconclusive battles outside Atlanta that dragged on and on. All seemed lost for the Union until Davis decided that he'd had enough of Johnston's delays, and relieved him in favor of Hood. Sherman was delighted with this news, as he knew that Hood could be counted on to engage in open battle, where Sherman could crush him. And that's more or less what happened. Sherman defeated Hood and seized Atlanta on September 2, after which the city burned to the ground. That was a huge morale booster for the Union (imagine how Lincoln felt after Sherman sent him a telegram reading "Atlanta is ours, and fairly won") and Lincoln secured re-election easily. But Hood's army was still more or less intact, and he wanted to redeem himself.

Hood's next move was to move northward towards Chattanooga, Tennessee and harass Sherman's supply lines, which he was reasonably effective at, so much so that Sherman sent General George H. Thomas to Nashville to organize the Union troops stationed in Tennessee and deal with Hood. Hood saw an opportunity to reclaim lost glory by defeating Thomas and taking Nashville. But to do that, he would first have to deal with John Schofield.

Schofield commanded the Army of the Ohio, roughly 27,000 men, who were stationed at the town of Spring Hill. Schofield's forces lay between Hood and Nashville, and Hood knew that if he let Schofield link up with Thomas, he would be badly outnumbered. He marched on Spring Hill, but despite numerous small skirmishes, miscommunication meant that Hood's army never managed to coordinate a major attack against Schofield. The result was that despite being encircled, Schofield's men simply got up and marched away on the night of November 29.

Franklin:

Hood woke the next day to find the Union forces gone. His response was to fly into a rage and demand to know what had happened. Once he found out where Schofield had gone, he marched on Franklin immediately.

There is quite a lot of debate surrounding Hood's decision to attack Schofield the way he did. Schofield's forces were well entrenched and had plenty of good artillery. Moreover, any attacking force had to cover several miles of open ground before reaching the Union lines, and the men manning those lines were battle-hardened veterans. Hood knew this, as did most of his army. And yet he ordered several waves of frontal assaults. Despite the overwhelming odds against the Confederates, they acquitted themselves rather well, even managing to breach the Union center at one point. But in the end they were totally repulsed. I am not aware of any scholarship arguing that he made this decision due to wounds from an earlier engagement, although it's not a bad theory, given that he had accumulated sufficient injuries that he had to be strapped onto his horse each day. I think he was probably growing desperate, and when we grow desperate we tend to fall back on what we know/are comfortable with, which in Hood's case was aggressiveness. Further, he probably didn't want to give Schofield any more time to slip away, which might have happened had Hood attempted a flanking maneuver.

Repercussions:

The battle was disastrous for the Confederates. Hood lost roughly 25% casualties, including Pat Cleburne, who had sufficient talent to be nicknamed "Stonewall of the West." Despite this crippling defeat, Hood still marched on Nashville, where the opposing sides made preparations for two weeks. In fact, Thomas took so long readying his attack that Grant actually sent a courier with orders to have Thomas relieved (although to be fair, a massive ice storm rolled in and paralyzed both forces for several days prior to the battle). But by the time the courier got there, the Union had won probably the most decisive victory of the war. In a series of overwhelming assaults, Thomas (bolstered by Schofield's Army of the Ohio) destroyed the weakened Army of Tennessee. The Franklin/Nashville campaign was the only military campaign during the whole Civil War that saw an entire army rendered totally hors de combat. What was left of Hood's army retreated south and never again took part in a major battle as a cohesive force.

I'm not sure how much back-story you're aware of, so I'll try and make it general. September 1864, things are going bad for the South. Lee's army is holed up in the siege of Petersburg, with his numbers getting thinner and Grant's numbers, facing him, getting bigger every month. The loss of Petersburg inevitably means the loss of Richmond. Sheridan was storming down the Shenandoah Valley against Jubal Early, outnumbering him two to one and about to cut off Richmond's major food supply. The North holds all major Confederate ports except Wilmington, North Carolina, crippling their economy and trade. Finally, on September 1, Hood's Army of Tennessee had been forced out of Atlanta, the biggest city in the Deep South, by Sherman, partly due to overwhelming numbers and superior leadership, and partly by Hood's habit of launching reckless and disjointed attacks.

John Bell Hood had been an excellent brigade and divisional commander in the East, at places like 2nd Manassas, Antietam, and Gettysburg, and his aggressive spirit served him well there. But he was not suited for higher command - the Peter Principle applies here; he was promoted one level, to corps command then another, to independent army command, above where he was effective. He was not a strategist or a long-term planner. I'll explain why below, but you are also correct in regarding his wounds: Hood had lost the use of one arm at Gettysburg, and lost his whole leg at Chickamauga. I personally think his battered condition is comparable to Marshal Ney from the last couple years of the Napoleonic Wars, but /u/DonaldFDraper might want to chime in there. It's not hard to think that Hood was never quite right, after that. Historians are divided on Hood's condition.

Sherman was far too strong, positioned in Atlanta, to attack directly; the Confederate high command's hope was that he would spread his forces out and leave them isolated, creating opportunities for victories. At first, Sherman seemed to do this. He was worried about his lines of supply and communication, mainly because Forrest was wreaking havoc back in Tennessee, control of which had long been lost to the South. Sherman sent George Thomas, one of the superior Union commanders, back to Nashville, and intended to send him some troops to clean up Forrest and deal with Hood if necessary.

Hood struck first, moving for the railroads and telegraph lines connecting Sherman back to the North. However, Sherman's detached troops were in strong positions (except for one unfortunate garrison with some black soldiers; six of them were killed after capture) and after a few skirmishes Hood to content himself with tearing up a little track before Sherman came rolling up with a massive force, driving the Army of Tennessee back into northern Georgia and then Alabama. It took Sherman eight days to repair the track, meaning the damage was minimal.

Hood decided that his best course of action was, as always, attack. He would attack into Tennessee, try to destroy whatever force Thomas could put forward, and then move into Kentucky, drawing troops as he went. Hood's strategy was basically to pull Sherman after him, away from the Deep South.

What Hood didn't realize was that Sherman had no intention of doing that. As far as Sherman was concerned, he had enough forces to deal with Hood, and then some - he had roughly 80,000 in and around Atlanta, 30,000 back in Tennessee, scattered, and more in Mississippi, Missouri, and Louisiana. He decided to detach two corps - IV Corps under Stanley and XXIII Corps under Schofield - from his main force under Schofield overall to keep an eye on Hood in Alabama, while he took the rest of his Army on the March to the Sea. Schofield was to report to Thomas, and it was his force of 30,000 that would fight Hood at Franklin.

Hood, for his part, had only 30,000 all together, and Sherman could toss off that amount like it was nothing. That should give you some idea of how insane this whole plan was. The Army of Tennessee, once the nemesis of the Union armies in the Western Theater, had been so beaten up that Sherman no longer considered it a major threat anymore, not even bothering to face them with his whole army. He basically left Thomas to scrape up an army from odds and ends to deal with a nuisance. This is the major reason why what happened next mattered so little, and seemed so futile.

Hood's plan was basically a pipe dream. He stood to accomplish nothing; he wasn't going to cut Sherman off, because Sherman was about to start burning and looting his way through Georgia, living off the land, while Hood's 30,000 were far too small of a force to actually accomplish anything against far bigger, if spread-out, Union forces. His only hope was to try and catch a smaller force and destroy it; even then, it's hard to be sure what he thought it would actually accomplish strategically.

So, the action. Hood tried to make a quick strike from northern Alabama into Tennessee, trying to get between Schofield's position and Thomas in Nashville. Due to a poor command structure, inexperienced leaders, and the nose-dive in morale of the Army of Tennessee, Schofield escaped, even when Hood should have had him. It enraged and frustrated Hood, and by the time he finally caught up with Schofield, on November 30, 1864, Schofield was entrenched with his back to the Tennessee River, at a town called Franklin.

It's hard to imagine what Hood felt like. The enemy had slipped him several times, and he had always failed to land his decisive blow, in front of Atlanta, or on the railroad, or a few days beforehand. Hood went all in. People debate why. Some say he was furious at his troops and commanders for failing him; I believe James McPherson argues for this. I have heard arguments that Hood felt he had no other choice besides retreat, and that he was determined, not angry. Differing accounts spring up, and like with any historical figure, there are as many opinions as there are people. I've read several variations, though off the top of my head I couldn't tell you who holds what opinion.

Essentially, Hood ordered a mass frontal assault of 20,000 men, against the advice of every one of his generals, and it went like such assaults in the Civil War, with very few exceptions, had gone. If you want a unit-by-unit play-by-play, I lack that knowledge; but there was some momentary success in the attack, mainly because a mix-up in Union orders had left one division under Wagner exposed, and they were routed by the attack. When the Southerners reached the main Union line, they made one small penetration, but Union reinforcements quickly closed up the gap. Elsewhere, the Rebels were stopped by heavy artillery fire, rifle volleys, obstacles and abatis, in some places making five or six attacks, each turned away in carnage. The losses among officers were staggeringly high, with six generals killed, seven wounded, and one captured.

One of the most notable deaths at Franklin was Patrick Cleburne, Irish-born and among the best generals on either side, shot while walking forward with his sword after his horse had been shot. Cleburne was one of the few generals who had held regular classes on military tactics and drill for his subordinates, had risen from private to major general, and had proposed enlisting blacks to make good the Confederacy's losses, to the shock of his peers. His loss was a massive blow to Southern morale, in particular the Army of Tennessee.

When the fighting cleared, the Southerners had suffered about 6,200 casualties from their force of 27,000, a loss rate of about 23%. Schofield's losses were in the area of 2,000, from roughly equal numbers, emphasizing how one-sided the battle was.

That night, on the 30th of November, Schofield withdrew from the battlefield and made his way to Nashville, arriving the next day. Hood followed - what else could he do? - and sat in front of Nashville, watching Thomas bring in more and more troops to deal with him. Sherman was already outside of Savannah,

What followed was the last major battle in the West, and Hood's Army of Tennessee went out with a whimper instead of a bang. Thomas had eventually assembled over 55,000 men from all parts of the Western Theater, while Forrest had joined Hood, bringing him back up to 30,000. It wasn't nearly enough. After waiting until he had all his ducks in a row, including more Union reinforcements from Mississippi and Missouri, Thomas struck Hood and in a two-day battle drove him from the field, with Thomas's troops suffering 3,000 losses to Hood's 6,000. The Union victory at Nashville, 15-16 December 1864, was probably the most complete victory they had ever won thus far, until Appomattox.

Hood's army staggered back into Mississippi, and if he hadn't presented a major threat before, Hood was finished now. In February he was relieved and the remnant of the Army of Tennessee transferred east to join forces trying to resist Sherman in North Carolina.

What was Franklin's impact? It was a heavy blow to an already half-dead Confederate Army of Tennessee, more losses that the Confederacy could not afford, and the battle was fought with no foresight and no good reason.

Franklin wasn't a game-changer. It was fought far too late in the war, especially in the West, to make a difference, and the stakes were low. Win or lose, Hood was not going to change the inevitable outcome by this point. His force was too small, the Union forces in the area too great, and the tactics at Franklin were horrible and badly thought out from the start. What scarred Franklin into history was the futility of it - the waste of good men, brave men, like Cleburne, for no strategic or tactical impact. By the time of Franklin, the war was already lost. They were smashing the deck chairs on the Titanic.