What theory was used by the United States government to justify intervention in many countries around the world during the Cold War?

The US had unsuccessfully intervened in Russia in the closing months of the Russian Revolution to try and support more conservative factions because of the imminent victory of Communist led forces. After Russia became Communist and reorganized as the USSR, its early years were spent looking inward, reorganizing its economy and solidifying its control over the vast expanse of Russian territory.

There was a brief "Red Scare" in the US and Western European countries after WWI because of the growth of Communist parties in Western society. But the brutality of the Communist party in securing its control reduced its popularity and its threat to the established order, even during the Great Depression.

During WWII, the USSR withstood the German assault, and then utilized its position to gain power over the Eastern European states on its borders. The West realized the USSR was now firmly in the grip of its Communist leaders, and that it had built a strong military force that was capable of expansion. Members of the US State Department realized they needed to formulate a policy to counter the expansion of Communism.

In 1946 and 1947, State Department diplomat George Kennan wrote the documents that gave the Containment Theory its outline. The US and its allies needed to "contain" the Communist threat within its existing borders, and to apply resistance whenever and wherever Communism seemed in danger of expanding. The policy documents were debated within foreign policy circles for the next several years.

It was the 1949 fall of the Nationalist government in China to Communist forces (and the American political fallout from that event) that led to the adoption of "Containment" as a near universal American policy mandate for the next 35 years and the resulting military intervention around the globe.