How did the Emancipation Proclamation proclaim freedom and instill hope in the slaves?
The Emancipation Proclamation did not actually free any enslaved people at the time it was issued. It proclaimed that all slaves in states "in rebellion against the United States" were free. People in rebellion against the United States would not free slaves because Lincoln declared them emancipated--in fact, defending the institution of slavery was the most important reason they had rebelled in the first place. Moreover, Lincoln did not free the slaves in the border states or even in some areas where Union troops had already taken over. He thought he lacked the constitutional authority to do this.
Despite this, the Proclamation was very important, in part because, as the question suggests, it gave hope to enslaved people. It redefined the war by ensuring that a Union victory would result in the destruction of slavery. This confirmed what many slaves had already recognized--if some people in the North saw the war as a war to preserve the Union, they had always seen it as a struggle for their freedom. Hundreds of thousands, in fact, had already flocked to Union lines for this reason. Another way the Proclamation gave hope to slaves in the South was to offer the opportunity to enlist in the United States armed forces. By the end of the war, more than 200,000 African-American men had taken advantage of this opportunity. Lincoln, in fact, credited black troops with helping turn the tide of the war. So the Emancipation Proclamation was of enormous importance, especially in redefining the meaning of the worst conflict in American history.