At the time, American policy makers subscribed to the "domino theory" where a communist country might cause nearby countries to go communist, either through inspiration or through subversion. A communist Cuba posed a threat to the Caribbean, Mexico, Central America, possibly even South America. It was the first Communist country in the Western Hemisphere and thus posed, for Americans, a frightening expansion of communist influence.
From a more practical standpoint, Cuba provided the Soviets with a staging base within spitting distance of US shores. This was the issue during the Cuban Missile Crisis, when the Soviets stationed nuclear missiles in Cuba. Those missiles could have hit American targets in minutes, leaving precious little time for Americans to respond.