The Cold War was a time period not of fighting, rather a clash of ideologies from existing countries. The two powerhouses at the time were the United States of America who stood for democracy and Russia who stood for communism. Their battles were never directly towards each other, rather it was indirect through proxy wars where the two powerhouses would try to help specific countries lean towards their standing.
During April 25th of 1959, Fidel Castro had met with Vice President Richard Nixon. Nixon questioned Castro as to why he went against the U.S imperialism. Castro would reply that he did it for the people of Cuba. Castro felt that by leading a revolution against the American government in Cuba, he would be able to give a better life to the people. “The Revolution taught the illiterate to read, provided healthcare to the poor, and reshaped the Cuban economy in spite of a crippling blockade” (Dawson, 243). In the moment, Castro felt that there needed to be a change in order to better the lives of the people, so he chose to go against the U.S to accomplish this.
https://nsarchive.gwu.edu/dc.html?doc=4329957-Document-10-Memorandum-of-Conversation-Rough
Alexander Dawson, Latin America Since Independence: A History with Primary Sources, (New York: Routledge, 2015)