I came away from today's lecture rather perplexed and did some fact checking of this statement. I used Wikipedia and Eric Foner's Give me Liberty volume 2 but found no reference to these missiles. So, I was wondering, how true is this statement really?
EDIT: Formatting
I think the most generous interpretation of your lecturer is that they confused two events into one:
One of the motivations for the USSR putting missiles in Cuba was US nuclear missile bases in Turkey, just over the Soviet border, pointing at the USSR. These particular missiles (Jupiters) were technically unable to be used as retaliatory weapons (they were too slow to launch and too vulnerable to even conventional attacks), so they were seen as only being justified as preemptive-strike weapons. This made the Soviets very uncomfortable, to have such things right on their doorstep.
The Cubans pushed for nuclear missiles to be located on their island as a guarantee that the US would not try to invade them again, as they had tried with the Bay of Pigs fiasco. The Cubans felt that the United States posed an existential threat against their country, and that only a direct deterrent with Soviet nuclear arms would prevent future US actions to try and topple their government with military means.
So one could charitably think that the lecturer transposed the missiles from Turkey to Florida and attached them to the US threat to Cuba. But it's a lot of transposition — it's misleading.
As an aside, the outcome of the Crisis was that the US agreed to remove the missiles from Turkey (an easy concession in a sense because the US felt they were not very safe either — they could be easily hijacked by the Turks, and this made them very uncomfortable), and for the US to pledge not to try and invade Cuba again. So both the USSR and Cuba sort of got what they want out of it, in the end, in exchange for the USSR agreeing to remove the missiles from Cuba.
As I noted in my other comment, the best source for understanding the motivations and thinking of the Soviets and Cubans is Mikoyan and Savranskaya's The Soviet Cuban Missile Crisis: Castro, Mikoyan, Kennedy, Khrushchev, and the Missiles of November (Stanford University Press, 2012). It revealed, among other things, that the Soviets were really not expecting the US to freak out as much as they did, that the Cubans were much bigger forces in arguing for the missiles (and for using them!) than we thought (that is, it was not just a bit of Soviet strategizing, but something the Soviets were somewhat badgered into by the Cubans), and that the "standard narrative" that JFK's efforts kept Soviet missiles and warheads out of Cuba is not exactly true — by the time of the crisis, the Soviets actually did have working missiles and hundreds of nuclear warheads in Cuba, and if the Crisis had gone poorly they likely would have been used against invading US forces and even strategically. So even at this late period there are still new things coming out about the Crisis, and it is worth being wary of US sources that seek to perpetuate the myth that it was a show of eyeball-to-eyeball brinksmanship. It was something more complex than that; the "eyeball-to-eyeball" narrative is one spread by Kennedy admin employees who sought to lionize JFK and his choices, and it misrepresents many key aspects. E.g. the original versions of it left out the fact that the US actually did give concessions to the USSR, like removing the Jupiter missiles. This changes the "lesson" a bit, from "be strong and never back down" to "sometimes reasonable compromise is a good idea."
Not really. Cuba is a scant 90 miles off the US coast. To put that in perspective, the range on the V2 rocket used by the Nazis during WW2 was about 200 miles. Essentially every missile system deployed in the United States was automatically within range of Cuba on account of Cuba being a lot closer than the USSR.
Now there is some argument to be made that the Soviets may have built up a nuclear presence in Cuba in response to the Bay of Pigs operation in which the US tried (and failed) to overthrow the Cuban government with an invasion backed by US logistics, support, and (in theory) air-power.
All of that said, the most widely accepted rational is that the Soviets put their missiles in as part of a response to or escalation of US missile systems in Europe.
As a primary source, Robert Kennedy's memoir "Thirteen Days" has some discussion of this, if memory serves, though I don't have it in front of me right now. You might also consult Zelikow et al's "Essence of Decision" which goes into extensive detail on the Cuban Missile Crisis.