So the U.S. Air Force has an official history division and their page on the origins of the Air Force has a plethora of information. Link

In particular, Herman Wolk's The Struggle for Air Force Independence, 1943-1947 goes into great detail on the moves that pushed the Air Force to increasingly become an independent branch from the Army.

In essence, the Army Air Corps, which then became subordinate to the Army Air Forces during WW2, already had a history of relative independence from the rest of the Army when war broke out.

The Army Air Forces also grew to millions of men during WW2, easily eclipsing the Marine Corps and being more akin to the Navy in size. With its own chain of command dealing with the intricacies of aerial warfare, the Army Air Forces essentially became a branch within a branch during the war.

Finally, the importance of air warfare was established in WW2. Achieving air superiority became a critical objective during every US operation. In addition, strategic bombing was an important part of strategy during the war and after the war, with nuclear weapons only deliverable by bombers at the time, aviation became more important than ever before.

Thus, the National Security Act of 1947 established an independent Air Force. No longer would the aviation command in charge of strategic bombing and air superiority respond to the Chief of Staff of the Army, who was likely to always be a ground based general.

In addition, the Air Force would now be solely responsible for the new spectrum of warfare that matured in WW2: domination of the skies. Army aviation would be limited to transport and logistics of ground troops, particular when helicopters came into being. The Navy and Marine aviation would remain focused on maritime operations and dominance at sea, which is separate from the Air Force focused on air superiority, strategic bombing, and later areas like airlift and space.