How did he convince the south to legitimize the emancipation proclamation?
Fairly early in the war Lincoln (actually one of his generals, but he endorsed the measure) took the position that human "property" in a rebellious state could be confiscated by Federal forces upon command, essentially as spoils of war. If the Federal government chose to employ these slaves as labor to help the army, or free them, that was the Federal Government's business.
The Emancipation Proclamation legally was viewed as an extension of this principle. It only applied to slaves in the states that were revolting (thus it didn't apply to Union slave states like West Virginia, Maryland, and Kentucky).
Of course the states in question didn't recognize Federal authority at the time, so in practice this just meant that if the South were to lose the war eventually, all their slaves would be freed. Effectively, it officially made the war a war to end slavery in the South. But it has often been remarked that the proclamation itself didn't free a single slave. At least not initially.