They didn't. The Confederacy couldn't care less about any orders given to them by Lincoln. The Emancipation Proclamation was a domestic power move in the Union. It was supposed to shore up support from abolitionists and make the war more about slave liberation and less about preserving the Union. Essentially, it was Lincoln acquiescing that the American Civil War was about slavery, not states' rights.

(Edit: A consequence I overlooked is that the Union explicitly coming out against slavery made the abolitionist powers of western Europe less likely to conduct trade with the Confederacy.)

Furthermore, it had immediate ramifications for the front. Militarily, the Emancipation Proclamation was written and delivered right at the beginning of a series of pushes south from the Union known as the Vicksburg Campaign. The goal of these maneuvers was to take full control of the Mississippi River, the lifeline of the Confederacy, and in the process to separate Texas, Louisiana and Arkansas from the eight eastern states. Supposedly (and in actuality) the Confederates would lose almost all access to trade if this campaign succeeded (and it did).

So, what does the Emancipation Proclamation have to do with the war effort? Well, it meant that any southern slave which reached the Union lines became free. This was immediately relevant for a few thousand slaves who were already under Union control, particularly in the Shenandoah Valley. However, it also incentivized slaves in the unoccupied Confederacy to flee and run northward.

We can tell that the Emancipation Proclamation was a political move moreso than a moral one because it suspiciously omitted Union-aligned states on the Confederate border from needing to free their slaves. This was a twofold military necessity; the border state economy needed to be kept intact to fuel the war effort, and if any of them joined the Confederacy in protest, it would be a disaster for the now-encircled Union troops to their south. So, they got off scot-free until the postwar ratification of the Thirteenth Amendment, which banned all forms of slavery nationwide.

If you have access to JSTOR, Union Strategy in Arkansas during the Vicksburg Campaign makes for some nice reading. Additionally (picture me crossing myself here, for I'm about to commit a grave sin), Wikipedia pages on U.S. history are generally very fleshed-out. As far as literature goes, literally anything by James McPherson will do - he's my favorite American Civil War historian. Battle Cry of Freedom is a very beefy book, but quite comprehensive, and it won a Pulitzer for a reason. For some more digestible (and free) reading, look to the University of Maryland's Documents from Freedom, a lineup of contemporary writings that relate to slavery and emancipation.