This is a pretty huge question. The first thing to keep in mind is that while part of the reason for mass emigration to the USA is related to Cuban government policy, another huge part of it is that the first world acts as a magnet for third world emigration everywhere, and the fact that the US has laws like the Cuban Adjustment Acts (AKA Wet Foot-Dry Foot) just encourages more people to make the trip. Remember, people also risk a lot crossing the inhospitable deserts which cover much of the US-Mexico border; this is not a purely Cuban phenomenon.

Cuban immigration spurred by political motives dates back to the 19th century, with important Cuban communities establishing themselves in New York City and parts of Florida (particularly Key West/Cayo Hueso). The second independence war (1895-1898) was organized and funded by Cuban exiles, particularly those in the US. In the first half of the 20th century these exile communities grew, paralleling analogous ones in Spain and Mexico. One example of this tendency towards exile is Fidel Castro himself, who went into exile in Mexico during the Batista dictatorship of the 1950s. Exiles gave support during the guerrilla war against Batista later in that same decade, sending weapons, other materials, and cash donations from the US. In short, exile has been a historical part of the Cuban experience for centuries at this point.

When Fidel Castro came to power he did so on a platform of the Revolution being ‘green like the palm trees’ (ie. Cuban, nationalist, and in accordance with tradition) and ‘never red’ (ie. Communist). His guerrilla movement, the 26th of July Movement (hereafter 26JM) had a broad base including middle class moderates like Huber Matos, who just wanted the restoration of the 1940 constitution which Batista had overthrown as a part of his coup in 1952. The Communist Party didn’t even support him at first, calling him a bourgeois who was trying to mislead 'the people' and accusing his attack against the Batista regime in 1953 of being a putschist attempt (ie. reminiscent of Hitler’s attempt at taking power in the early 1920s; yes, they compared him to Hitler). The movement also channeled a desire to return to the roots of Cuban nationality as a political project, in particular the political writings of Jose Marti, in order to ‘restore’ the Cuban Republic to how it was meant to be (ie. eliminate the huge issues with corruption, gangster like tactics in politics, and nepotism in government during the late 1940s and early 1950s). So when Fidel came to power in 1959 (first behind the scenes, then outright) it was difficult for him to try to turn this Revolution, which had been about restoring a Liberal Democracy, into a Proletarian Revolution against the bourgeoisie.

Right after Fidel came to power the Revolution started by seizing the property of Batista and his cronies, arresting the most brutal criminals in the regime, and killing the worst offenders (and allegedly innocents as well) in summary trials which ended in firing squads in front of a wall (paredón). Many seemingly legitimate businesses, but which had ties to the mob or were just considered corrupting influences (houses of prostitution, casinos, etc.) were also seized. This created the first wave of enemies/exiles of the Revolution, often called batistianos (Batista supporters), even though some were simply fleeing because they lost their businesses related to prostitution, gambling, or other vices frowned upon by the new government. That´s 1959.

Then came the exiles from the period between 1960 and 1961 (which includes my family) who fled because Fidel´s rhetoric was more and more openly Communist and the seizures of property had begun to embrace large landed properties (for land redistribution) among other issues. Because this hurt some of the same people who had supported the Revolution when it was in its Nationalist Radical Reform stage it caused people unaffiliated with the Batista regime to leave in the tens of thousands.

After the tensions between the US and Cuba over the Bay of Pigs invasion (financed, organized, and instigated by the US government) there was a cut-off in immigration abroad, which slowed to a trickle. The Cuban Adjustment Act, passed in 1966, facilitated Cuban immigrants turning into Legal Permanent Residents.

During this period, from 1962-1979 a series of Cuban government policies hurt a lot of people and caused a lot of resentment. The UMAP labor camps for LGBT people and those influenced by so-called bourgeois culture (ie. hippies, those who liked the Beatles, etc.) which lasted between 1965-1968 are one of the biggest stains on the Revolution´s record. These were not death camps, but they were reeducation camps, which told homosexual men that they weren´t real men but insisting that doing men´s work (cutting sugar cane, doing other agricultural labor, all for free) would make them men. The sign above the camp gates had that written in big letters: El trabajo les hará hombres (work will make you men). Press censorship started getting bad by the end of the 1960s, the government took control of the arts, it started suppressing the work of those it didn´t like, harassing and jailing those who criticized it. Guillermo Cabrera Infante, a Cuban author who went into exile around this time, has often talked about how he had gay men sleeping on the floor of his apartment seeking shelter because he was an editor at an important political magazine around that time and they hoped the government wouldn´t go so far as to breaking into his home. The 1971 Congress on Education and Culture began the so-called Quinquenio Gris (Grey Half Decade) when it became official government policy that homosexuals could not be teachers, doctors, or in any other way in a position to influence and “corrupt” Cuban youth (ie. the old trope that gay people “recruit” others). The arts became an extension of politics, the religious were barred from attaining positions in the government, etc. Needless to say, outright political opponents of the regime had it even worse.

By 1980 Cuba was a pressure cooker ready to blow. This is when several individuals broke into the Peruvian Embassy and asked for asylum. When the Peruvian government refused to return them to the Cuban government the latter rescinded the embassy´s security detail and let anyone who wanted to enter the embassy do so (officially this was to face trial for murder, as in their attempt to enter the embassy they killed the security guard, Pedro Ortiz, who worked for the Interior Ministry and was assigned to guard the Peruvian Embassy). The short version of this is, one thing led to another and Fidel said anyone who wanted to leave Cuba could do so. This was the so-called Mariel Boatlift, so named due to it being organized between ports in Florida and the Cuban port of Mariel. Many of those who decided to leave were called escoria (trash) by government organized harassment mobs, who were doing what were called actos de repudio (acts of repudiation), which included insulting and throwing refuse and eggs at neighbors of yours who were leaving. Lists would be taken of who participated and if you failed to do so your absence would be noted (to your detriment).

Many people have never forgotten that humiliation or the government instigators behind much of it, for good reason.

The 1980s were a period of relative abundance in Cuba so immigration for economic reasons diminished while political (here I´m included persecuted LGBT and religious people in addition to Liberals and Conservatives) immigration continued.

The huge wave of immigration in the 1990s and since has more to do with economics than anything else, though the political situation has not helped. The 1990s coincides with the collapse of the USSR and the rest of the socialist bloc in Eastern Europe, which represented the loss of over 50% of all Cuban trade. People went hungry, temporary blindness due to vitamin deficiencies occurred, among other issues. It was a terrible period popularly known as el periodo especial (´the special period´). Understandably, people left however they could.

Again, Fidel allowed those who wanted to leave to go with whatever would take them, which now included boats made out of anything that could be retrofitted to be seaworthy.

Compounding this was that the Cuban Adjustment Act was amended in 1995 to allow for anyone from Cuba who merely stepped onto dry, American soil to become a candidate for parole as immigrants and become legal permanent residents within a year.

I can´t really go into more recent years due to the 20-year-rule, but I´ll summarize it by saying that while things have gotten better, the economic and political motives for leaving are still present.

In addition, we have to understand that this is all happening in the context of a world-wide tendency towards immigration from the third world to the first world for economic reasons. Cubans aren´t special for trying to come to the USA on rickety rafts. This happens in the Mediterranean as well, though that is likely less dangerous because immigration from North Africa to Southern Europe doesn´t have to deal with the Gulf Stream or other tropical storms.

Bibliography:

Felix Masud-Piloto: From Welcomed Exiles to Illegal Immigrants: Cuban Immigration to the US from 1959-1995.

Jorge I. Dominguez: Cuba since 1959 in The Cambridge History of Latin America: Volume 7: Latin America since 1930: Mexico, Central America and the Caribbean, Leslie Bethell (ed.).

Lars Schoultz: That Infernal Little Cuban Republic: The United States and the Cuban Revolution.

WARNING: FOLLOWING CITED SOURCE IS A POLITICALLY BIASED DOCUMENTARY. IT IS VALUABLE THANKS TO SEVERAL IMPORTANT INTERVIEWS WITH KEY FIGURES BUT SHOULD BE APPROACHED CRITICALLY.

Conducta Impropia (Inappropriate Conduct, 1984). Dir. Nestor Almendros.