My sources are the many references to this fact I have heard throughout my life, but if you need something to read --
World Book
History of Hispanic American immigration
The Hispanic presence in what is now the United States actually began before the country existed. Spanish explorers established colonies in what would become the Southeast and Southwest regions of the United States.
Cuban immigration
Cuban immigration to the United States picked up sharply during the late 1950's, as a result of increasing political turmoil in Cuba. Until the mid-1950's, only a few thousand Cubans came to the United States each year. But during the late 1950's and early 1960's, the number of Cuban immigrants increased dramatically. In 1959, Cuban revolutionary Fidel Castro came to power. He announced the restructuring of Cuban society. Many middle- and upper-class Cubans found Castro's plans threatening to their way of life. Between 1959 and late 1962, about 200,000 anti-Castro Cubans immigrated to the United States.
In October 1962, commercial air flights between Cuba and the United States were suspended. Nonetheless, about 50,000 Cubans entered the United States between late 1962 and 1965. Many of these people sailed secretly from Cuba in small boats, some of which were apprehended by the Cuban navy before they reached the United States. In 1965, the United States and Cuban governments agreed to set up an airlift between Cuba and Miami. The airlift brought about 250,000 Cubans into the United States between 1966 and 1973.
Until 1986, the United States admitted most Cuban immigrants who claimed to be seeking political asylum. Many of the first Cubans to flee in the early 1960's were from wealthy families and were well educated. The U.S. government granted asylum to these people and offered federal help to qualified applicants in finding homes and in making job contacts. Most later Cuban immigrants were relatives of the first group or were poor people looking for work.