Communism and Fidel Castro
The nation was left with a number of warring factions and it was always under a state of turmoil. The Communist Party grew in size with the help of military and famous Cuban leaders like Fulgencio Batista. The country was ravaged by coups and overthrows by Army. The money laundering, drug and flesh trade mafias sprung up. The late fifties saw the emergence of Fidel Castro and he became the Prime Minister of the country in Feb 1959. His pro Russian policies earned him the ire of the US.
The US even today has trade embargo against Cuba. If you are a US national and plan to visit Cuba, do so after proper approval from the US Government, else you may land into prison and need to cough up a hefty fine too.