Leopards Ate My Face

There's more context to it. When Castro first took over Cuba, the first wave of emigrants to flee the country was referred to as "the Golden exile." These were people who already had wealth and/or connections to the previous Cuban political regime. They feared losing their status and privileges under communism, so when they left and settled in other places, such as south Florida, they brought their beliefs and attitudes with them.

From the linked Wikipedia page:

By the middle of 1959 various new policies had affected Cuban life such as the redistribution of property, nationalization of religious and private schools, and the banning of racially exclusive social clubs. Those that began to leave the island were driven by them being negatively affected by new economic policies, their distaste with new national public schools, or anxiety over government supported racial integration. The government would quickly label exiles who left as "racists", discouraging Afro-Cubans to also emigrate. These conditions caused the majority of those who emigrated to be either upper or middle class, white, and catholic.

As a result, a large number of Cuban exiles that came to the United States aligned themselves with the Republican party. To this day, the connection remains (and is encouraged by politicians that can benefit from it, as another commented noted.)