How does the Emancipation Proclamation relate to the Declaration of Independence?

It doesn't. Technically there's only one similarity: both freed people on paper but couldn't actually free them until the successful win of a war.

Remember: Lincoln's EP only freed the slaves in rebellion. Any state still in the union - like Kentucky and Maryland - those slaves remained in slavery. Lincoln freed people he had no direct control over (at the time) and left in bondage those he could have freed (except it would have been unconstitutional.)

The declaration of independence states: We hold these truths to be self-evident that all men are created equal. Slavery was a stark contradiction to this professed belief. The Emancipation Proclamation declared the freedom of men from slavery and was the beginning steps to correcting this contradiction.