Why did the united states and soviet union enter into the cold war?

The underlying causes of the Cold War came from dramatically different worldviews held by the United States of America and the Union of Soviet Socialist Republics. The USA was committed to capitalism and democratic institutions of government. The USSR was committed to communism and imposed authoritarian government. The Cold War was mostly a tension between these worldviews.

Additional explanation:

There also were immediate issues in the aftermath of World War II that drove the USA and USSR from being allies to being rivals. The USA had atomic weapons and the USSR did not. (The US would not share that technology with the Soviets.) By 1949, the USSR developed and tested the first of its own atomic weapons, and the Cold War rivalry became a nuclear rivalry. The two countries were also at odds over how governments in Europe would be reconstituted after World War II. The USA wanted free and fair elections and democratic countries to develop, whereas the Soviet Union was looking for governments in Eastern Europe that would align with its communist system and provide a security buffer against what it saw as capitalist imperialism.

The US got involved in the Cold War because it believed that communism was a threat to the stability and freedom of the world. It believed that the Soviet Union would do whatever it could to spread its ideology to as much of the world as was possible. The US believed that this would be bad both for the US and for the world.