The post-World War II spread of communism can be attributed to a number of different factors. Firstly, the Soviet Union's ultimate success in driving Axis forces out of its territory naturally led to Soviet armies crossing into the nations of Eastern Europe. This included nations that had been conquered by Nazi Germany - Poland and Czechoslovakia - and nations that had chosen to become the allies of Nazi Germany - Bulgaria, Hungary, Romania. Following the end of World War II in Europe in May, 1945, with Soviet troops occupying these countries, as well as the Eastern half of Germany, the Soviet dictator, Joseph Stalin, made certain that communist governments took control of these nations, so that all of Eastern Europe became communist.
In Asia, beginning in the 1920s, there had been an ongoing civil war in China between Chinese Communists, led by Mao Zedong, and the anti-communist government, led by Chiang Kai-shek. This civil war was interrupted by the invasion of China by Imperial Japan in 1937. But once Japan was defeated and forced to surrender to the U.S. and its allies in September, 1945, the civil war in China between communists and anti-communists broke out once more. By 1949, only four years after the end of World War II, Mao Zedong's communist forces had gained complete control over continental China (Chiang Kai-shek's anti-communist government was forced to retreat to the island of Taiwan, located off the coast of China).
Thus, because of the outcome of World War II in Eastern Europe and the outcome of the civil war in China, only a few years after the end of World War II communism had spread far beyond the borders of the Soviet Union.
Conflict between those who believed in communism and those who opposed communism - essentially those who practiced capitalism - was guaranteed by the basic ideology of communism. Communist ideology argued that humanity could only achieve perfect justice and happiness when capitalism was destroyed and replaced by the communist system in which there would be no private property - everything would b e shared equally by everyone, so that there would be no divide between rich and poor. The government of the Soviet Union, the puppet communist governments of Eastern Europe controlled by the Soviet Union, and the Chinese government under Mao Zedong were all, more or less, dedicated to replacing the capitalist system throughout the rest of the world through the violent destruction of non-communist governments everywhere.
This tied in with a third factor in the immediate post-World War II world. A combination of the promotion of the ideals of liberty by the United States during its war against the Axis powers (Nazi Germany, Fascist Italy, and Imperial Japan) and the damage caused by the war to the major European nations that possessed overseas empire - particularly France and Great Britain, but also the Netherlands - resulted in the achievement of independence by a large number of nations in the 'Third World / Developing World', particularly in Africa and Asia. While some colonies were given their independence voluntarily by their European rulers - Great Britain voluntarily handed over control in India-Pakistan - in many African and Asian colonies, the native peoples had to fight wars to win their independence (not unlike our own thirteen colonies in 1776). These anti-colonial movements needed money and weapons to wage their wars against the forces of European nations such as France. The Soviet Union and Communist China saw this as an opportunity to spread communism and enthusiastically provided anti-colonial movements in the 'Third World / Developing World' with the money and weapons they needed, along with advisers to teach their troops how to wage war.
It's easy to see why capitalist nations, particularly the Western democracies of Europe and the U.S., were upset about the spread of communism. As explained above, communism was dedicated to their destruction. In only a few years after the end of World War II all of Eastern Europe and China had become communist and there were communists seeking to take control of former European colonies in Africa and Asia. It seemed to the western capitalist democracies that, unless it was stopped, communism would achieve its goal of taking over the world.