Why would the thirteenth amendment help with ending the war?

I’m going to be a little cheeky in posting this answer, as I never studied American history in school and so almost all of my comments have been taken from other sites as opposed to my own knowledge.

The first thing I must point out is that whilst you’ve asked this about the Lincoln movie, this is obviously a question related to the actual politics of the time and thus is a very deep and debateable question. Having done a little research online on it, here are some of the things I’ve come up with.

Summarising from Think Quest (http://library.thinkquest.org/J0112391/myth_8.htm):

In July, 1862, the war was going badly for the North, Lincoln decided to try to free the South's slaves to help the war effort, but he waited until the Union had a military victory. This was because Secretary of State William H. Seward feared that with recent Northern military defeats the proclamation might be looked on as a "last shriek on the retreat.", i.e. an act of desperation from the losing side. However, on September 17, 1862, in the Battle of Antietam, the Union stopped Confederate General Robert E. Lee's attack on the North in a morale boosting victory. Even though Antietam wasn't prearranged, it was good enough for Lincoln's purpose and he passed the Emancipation Proclamation. It helped the Union by reinforcing the North's war effort and weakening the South's. The South was hurt by the Emancipation Proclamation because it discouraged France and Britain from entering the war. Those two countries depended on the Confederacy, or South, to supply them with cotton, so the South hoped that they would fight on their side. Most French and British citizens were against slavery, though, so when the proclamation made the war a fight against slavery, France and Britain gave their support to the Union.

Summarising from How Stuff Works (http://history.howstuffworks.com/historical-events/lincoln-emancipation-proclamation.htm):

While it didn't technically set anyone free, the Proclamation was part of Lincoln's strategy to demoralize the South, and it worked. Poorer Southern whites resented that they were now fighting a war to protect wealthy plantation owners who were desperate to hold onto their "property." And as word of the Proclamation spread, slaves left those plantations en masse. Their exodus even helped turn the tide in the siege of Vicksburg, a vital Union win. Additionally, France and England, which had been secretly helping the South, could not officially recognize a country that still enslaved other human beings. Europe also could not provoke a country that, according to the Emancipation Proclamation, was now fighting slavery.

Summarising from Civil War.org (http://www.civilwar.org/education/history/emancipation-150/10-facts.html):

Fact #6: The Emancipation Proclamation changed the focus of the war. Up until September 1862, the main focus of the war had been to preserve the Union. With the issuance of the Emancipation Proclamation freedom for slaves now became a legitimate war aim.

Fact #8: The Emancipation Proclamation paved the way for African-Americans to fight for their freedom. Lincoln declared in the Proclamation that African-Americans of “suitable condition, would be received into the armed service of the United States.” Five months after the Proclamation took effect; the War Department of the United States issued General Orders No. 143, establishing the United States Colored Troops (USCT). By the end of the war, over 200,000 African-Americans would serve in the Union army and navy.

Fact #10: Lincoln considered the Emancipation Proclamation the crowning achievement of his presidency. Heralded as the savior of the Union, President Lincoln actually considered the Emancipation Proclamation to be the most important aspect of his legacy. “I never, in my life, felt more certain that I was doing right, than I do in signing this paper,” he declared. “If my name ever goes into history it will be for this act, and my whole soul is in it."

Therefore, having summarised a few different sources of material, it seems the main reasons for Lincoln’s actions were:

1. He genuinely believed it was the right thing to do.
2. More importantly, he hoped to stop Britain and France giving any aid/support to the South.
3. He hoped to turn the poorer South army against their “masters” by showing them they were effectively fighting to allow their rich masters to retain their slave workforce.
4. He hoped to increase the size of the North’s army.

As stated at the top of the answer, I got this from a little research. I never studied this in school and I'm not American, so if anyone has any issues with any of the info in the answer, do please comment! .