This earlier answer by /u/TheLatexCondor should give you some insight. If the subject interests you, then I recommend you seek out Bruce Levine's book Confederate Emancipation: Southern Plans to Free and Arm Slaves During the American Civil War (Oxford University Press, 2006) that details the whole saga.

To piggyback on what TheLatexCondor already wrote (and I encourage you to read that answer in full), context is key. From the start of the war, the Confederates attempted to maintain the status quo on the slavery front, including in their diplomatic relations with Britain and France. (Incidentally, slavery was the only "state right" that the Confederacy stood by throughout the war, having reversed course on most of the others.)

The "arm the slaves" plans were first seriously proposed in the Confederacy in 1863, followed by the "emancipate the slaves" plan in January 1864. In both cases, these schemes were first proposed quietly, within the Confederate government. And, in both cases, they were quite aggressively opposed. When the latter suggestion first made its way to Jefferson Davis's ear from its origins with a commander in the Army of Tennessee, not only did Davis reject the proposal out of hand, but he prohibited any further discussion of the matter in the army - at all.

But the Confederacy's situation and ability to continue to fight the war deteriorated throughout 1864. To offer some more context:

In April 1864, the U.S. Senate had passed the 13th Amendment that would end slavery under an "immediate emancipation" plan. It got a large majority in the House, too, but fell short of the 2/3 majority needed by about a dozen votes.

In November 1864, Lincoln won re-election, and the Republicans won an additional 40 seats in the House, for an overwhelming super-majority. Lincoln had already endorsed the 13th Amendment, and the Republicans were poised to try to pass it again just as soon as the new session of Congress was inaugurated in January 1865. The 13th Amendment's "immediate emancipation" plan would end slavery everywhere in the Confederacy, should the U.S. win the war.

Jefferson Davis did not reverse course on the issue until December 27, 1864, the date he authorized Kenner to embark on his diplomatic mission. By that time, Confederate troops were deserting at an unsustainable pace - for a lot of the South, the war was essentially over the day Lincoln was re-elected. There was no stomach for four more years of war, which is what it would almost certainly take before the Republicans would lose either Congress or the White House.

To keep the war going, the Confederate military needed reinforcements. This is what convinced Davis, and what caused him to reverse his position. His proposal was for 250,000 black troops to be ready to fight for the Confederacy by the end of 1865. In exchange, these troops would be emancipated at the end of their military service.

But by the time actual formal debates on this proposal occurred in Confederate Congress in February 1865, most of the Southern armies had already collapsed. In the East, Robert E. Lee and his Army of Northern Virginia were trapped near Richmond and couldn't get out. His troops were deserting, and they had no way of replacing them. The rest of the Eastern armies had so few troops that they were re-organized into the broader "Army of the South", which couldn't even muster together 25,000 troops for one battle, which they lost. Desertion continued and they never fought again.

It was in this context that Jefferson Davis sent Duncan Kenner on a mission to Europe to make the "gradual emancipation" proposal, to counter the U.S.'s "immediate emancipation" designs under the 13th Amendment. Things were so bad for the Confederacy that it took Kenner a month before he was even able to set sail on his mission. The U.S. Navy had all the Confederate ports blockaded, so Kenner had to sneak into the Union and board a ship in New York City. The 13th Amendment passed the U.S. House just a few days after he left, so the same week that Kenner was making his "gradual emancipation" proposal to try to win over European support, the Europeans were aware of it in the context of the "immediate emancipation" that the Union was moving forward with.

In other words, while the Confederate plan was a concession when compared to their earlier position, it was still an effort to try to thwart the fast-approaching end of slavery, at least for a couple decades or so longer. Their proposal would also leave them with the possibility of politicking their way out of the promise later on. Even with the new concession, it was still better for slavery than defeat by an "immediate emancipationist" North, as had now come into existence, where there was no room for bargain anymore on the slavery issue.

As Levine's book details, once these proposals became public--first, the "arm the slaves" proposal in October 1864, then the "gradually emancipate the slaves" plan in January 1865--there was quite fierce public debate about them in the Southern newspapers and in the halls of Confederate Congress. A few newspapers, including the leading voice the Richmond Enquirer, lent their support. But most others were deeply critical, often re-hashing pre-war arguments with Northern abolitionists -- emancipation is beyond federal authority; it's a "state's right" to end slavery; ending slavery would violate slaveholders' property rights, etc. But another argument was also made, which can be summarized as, "If we give up slavery, then the Union has already won, so why bother to keep fighting this war."

That message seems to be the one to have resonated. Or, at least, the proposal did nothing to reinvigorate the war effort, as it was designed to do. The promise of 250,000 black troops by year's end did nothing to boost white troop morale and keep the war going long enough for those reinforcements to arrive. If anything, it did the opposite, because the rate of desertion got worse throughout the first half of 1865, not better. By mid-year, there weren't enough troops to keep fighting, and surrender had begun.

So, to borrow again from what TheLatexCondor wrote, the whole Kenner episode was delusional and desperate. Lee was just weeks away from surrendering when Kenner set off for Europe. And then when the Confederate government made the proposal public, the reception was ice cold. Certainly, it was far from the morale-boosting, momentum-shifter that the Confederate leadership hoped it would be. Rather, it may have expedited a process already going on, because it got Confederate troops, newspaper editors, and politicians to begin to openly question what the war was about if not for slavery, as Levine's book details.

"What did we go to war for, if not to protect our property?" wrote former Confederate Secretary of State Robert M.T. Hunter in an editorial, by then representing Virginia in the Confederate Senate. In the Confederate House, Rep. Robert P. Dick of North Carolina gave a widely-published speech denouncing the emancipation proposal as "opposite to all the sentiments and principles which have heretofore governed the Southern people".

The war ended shortly thereafter.

As for the arguments that Confederates made to defend these "arm and emancipate the slaves" proposals, they were mostly along the lines of fear of Northern control. That is, if the South lost, there would be retribution/legal ramifications against former Confederates by "Yankees" and the newly "freed Negroes". There would not just be prison sentences and executions of Confederate political and military leaders, but the North would confiscate the land and homes of white Confederate sympathizers. That land was then going to be given to their newly-freed black neighbors, and/or Northerners who would move South to cash in.

Gov. Henry W. Allen of Louisiana gave a typical speech in January 1865 on these grounds. He warned that if the Confederacy were to lose the war, "your negroes will be made your equals, your lands will be declared confiscate, and you will become the slaves of the very hirelings who are now waging war upon you".

Therefore, the wisest course of action was to move forward with a more gradual emancipation plan. The Confederacy could use those freed people to achieve independence, and then, later, control the timetable of emancipation. But again, the public debate was fierce and not very welcoming of these new proposals. Ultimately, the Southern reaction to these plans was continue the path toward surrender, rather than to see any of these new plans through.

Of course, the fears that the Southern proponents of these plans warned of never really came true. There was no race war. There were no mass Confederate executions. Nothing like that. And because nothing like that happened, the disgraced Confederate leadership who wrote about the conflict in the post-war years were usually content to sweep the whole episode under the rug.

Of particular note is Jefferson Davis's memoirs, he being the one who authorized Kenner's mission to Europe. In Davis's 1000+ page The Rise and Fall of the Confederate Government, he doesn't make mention at all of the proposal, nor of Kenner's mission. But he does spend two chapters criticizing the Emancipation Proclamation as illegal and an overreach of federal authority, while also defending Southern slavery itself as "confessedly the mildest and most humane of all institutions to which the name 'slavery' has ever been applied". Clearly, Davis had not authorized the Kenner mission because he'd had a change of heart on the morality of slavery.