It's a saying that has roots in the immediate aftermath of WW2 and the evolution of the Cold War. It became a more popular phrase as time progressed and there was ever growing rifts between the 'free world' (1st) world, and the communist 2nd world.

The notion that Americas national purpose was "to lead the free world" emerged gradually between 1947 and 1950. From 1950 on, it became refined; the image of the United States as Leader of the Free World served as the central trope in U.S foreign-policy discourse at lease until the collapse of the Soviet Union in 1991.

Fousek, John, To lead the free world: American nationalism and the cultural roots of the Cold War (Chapel Hill, 2000) p.130