Short form: because the Proclamation listed specific areas and included the phrase "are and henceforth shall be free", those slaves were and remained free as the United States forces advanced.

The US Archives has the text here

The wording is significant. It didn't say "slaves would be free in states still in rebellion". It named ten specific states, two with territorial exceptions where the United States had extensive control:

Arkansas, Texas, Louisiana, (except the Parishes of St. Bernard, Plaquemines, Jefferson, St. John, St. Charles, St. James Ascension, Assumption, Terrebonne, Lafourche, St. Mary, St. Martin, and Orleans, including the City of New Orleans) Mississippi, Alabama, Florida, Georgia, South Carolina, North Carolina, and Virginia, (except the forty-eight counties designated as West Virginia, and also the counties of Berkley, Accomac, Northampton, Elizabeth City, York, Princess Ann, and Norfolk, including the cities of Norfolk and Portsmouth[)], and which excepted parts, are for the present, left precisely as if this proclamation were not issued.

And it also states the result:

I do order and declare that all persons held as slaves within said designated States, and parts of States, are, and henceforward shall be free; and that the Executive government of the United States, including the military and naval authorities thereof, will recognize and maintain the freedom of said persons.

So the answer to the question: because of the areas plus "are and henceforth", as United States troops advanced into the specific geographic bounds, the persons held in slavery in those places were deemed by the Executive Branch to be free.

Given the recent history of emancipation, nothing else could have happened. General Butler had already deemed that slaves coming under his army's control would be considered "contraband of war". The US Confiscation Act of 1861 legalized confiscation of slaves as war material, though it had little effect. More effective was the 1862 Act Prohibiting the Return of Slaves (by the armed forces). Contrabands were already contributing as laborers and as soldiers. While abolition was a rather extreme act at the moment, returning captured black people to slavery by then would probably have sparked massive outcries and the loss of a lot of support that the United States considered valuable.

And it would have been silly to say that suggested alternative, that advancing US troops re-enslaved people. That would have made the Emancipation Proclamation utterly useless, and that was a concern for Lincoln, who stated that he didn't want a useless proclamation: "It would be like the Pope's Bull against the comet". It would have also ruined Lincon's purpose: he justified it as a wartime act against the productive capacity of the Confederacy, giving an incentive for the support of black people against the Confederacy and for the United States. Having re-enslavement happen would have removed any incentive.