Short version. . . we swiped Cuba from the Spanish in the Spanish-American War in 1902, and imposed a provision in the constitution of "independent" Cuba that the U.S. could "intervene" and "supervise" whenever absolutely necessary or we felt like it.
Over the decades, Cuba became a country of dirt-poor sugarcane and coca farmers and glitzy hotels on the coast with gambling, hookers, and blow for Yanks who were far too good Christians to do that stuff at home. Cuba was run for cheap sugar and easy access to the aforementioned GH&B until it decide to get really independent in 1959. Having experienced the U.S. version of capitalism, Cuba chose to be communist in name, though it doesn't really matter what tag you hang on a thugocracy. So we tried stupid interventions for a few years, which drove Cuba into the arms of the Soviets for self-defence. After the Soviet Union collapsed, Cuba fell on hard times, though they still cling to the "Communist" tag.
It's ultimate fate will be that of any other tropical island off the shore of a military and economic superpower. If they had sense enough to pound sand down a rathole, they'd go all in for GH&B, and use the proceeds to provide a decent life for their people (as well as mega-riches for the connected). But they don't, so what're ya gonna do?