In 1979, Fidel Castro began allowing Cuban Americans to visit Cuba, with over 100,000 visiting and spending around $100 million dollars (collectively). Many spoke freely about how wonderful the U.S. was.

As a result, boat hijackings began, as people tried to flee to the U.S. In April 1980, a group of Cubans hijacked a bus and drove it through the gates of the Peruvian embassy, killing a Cuban policeman in the process. At that time, citizens were not allowed to enter foreign embassies freely to claim political asylum. Castro wanted the hijackers turned over to the authorities and the embassy refused, so Castro removed the police guarding the embassy and 11,000 Cubans flooded the embassy grounds.

In response, Castro opened up the ports to anybody who wished to leave Cuba and as long as somebody would pick them up. From August to September, 125,000 Cubans left for the U.S. Cuban exiles in the U.S. paid boat operators in the U.S. to pick up relatives from Cuba. Eventually the Coast Guard and then the U.S. Navy participated in the boatlift.

As for Castro releasing prisoners and mental patients, yes that did happen. Castro did so in order to get rid of them and to harm the image of the refugees.

Keep in mind, this was the second wave of Cuban refugees. The first wave occurred just after the revolution and was comprised of the middle and upper classes of Cuban elites. By the time of the Mariel Boatlift, this first wave had settled comfortably in the U.S. The second wave was primarily comprised of people of color who were poor and uneducated. The first wave was motivated by political reasons, the second wave was political and primarily economic. Soon after the boatlift, the Cuban American community turned against the "marielitos" because they believed they were tarnishing their image.

By September, both Cuba and the U.S. agreed to end the boatlift.