Both the USAF and the CIA flew several overflights of Cuba with Lockheed U-2 spyplanes during both the Eisenhower and Kennedy administrations. The rationale for doing so was intertwined with efforts to both topple Castro and gauge the actual status of military forces within Cuba.
The first overflights were part of the failed Bay of Pigs invasion. The CIA had asked the NSC to approve of overflights for the purposes o gaining intelligence of Castro's military forces and his capabilities. Operation Kick Off began in October 1960 and continued with Operation Green Eyes in November and December 1960. Kennedy continued these overflights with the impending Bay of Pigs invasion with Operations Long Green and Flip Top. The intelligence gleaned from these overflights though was not enough to make the invasion a success and it proved to be a costly fiasco.
Ironically, the failed invasion as well as the persistent overflights created the very Soviet build-up on the island Washington feared. Castro had been moving to an openly pro-Soviet stance before the Bay of Pigs, but there he had little in the way of actual Soviet military equipment on the island in April 1961. The Bay of Pigs provided both a justification and an impetus for this type of assistance. The Soviets began moving in more sophisticated weaponry onto the island such as surface to air missiles (SAMs), MiG-21s, as well as conventional equipment like tanks. The Soviet build-up prompted a similar increase in U-2 flights, codenamed Nimbus, that averaged around two overflights per month starting in the spring of 1962. The Nimbus flights tended to be cautious "in and out" missions that minimized the time the U-2s could be in danger of being shot down by SAMs. But American penetrations of Cuban airspace grew more aggressive after a brief pause as intelligence rumors of Soviet surface to surface missiles were on the island surfaced. These missiles could have a nuclear payload and were the ones discovered by U-2 flights in mid October 1962.
So the US was not innocently flying around the island and just stumbled on the Soviet missiles. The discovery of the missiles was part of a larger and deliberate reconnaissance program to gauge the exact nature of the Soviet military build-up and Cuban military capabilities. The US intelligence agencies suspected the presence of such weapons or the Soviet development of the island into a larger nuclear base, but they had no direct evidence. Aerial reconnaissance was a risky but effective means to dispel these unknowns, or in the case of the October flights, confirm them.
The US under both CIA and USAF direction overflew numerous Warsaw Pact and USSR aligned nations, along with other conflict zones in the 1950's and 60's using U-2 aircraft. Everywhere from Poland and East Germany, to Cyprus, Egypt, and Indonesia received overflights to gather intelligence and take detailed photos of sites of interest. And of course many overflights or border skimmings' of the Soviet Union itself. It was initially thought that Soviet radars would be unable to track the flights, and when this failed to be the case that its very high altitude would keep it safe from Soviet interceptors and missiles. This proved to be the case initially, however as the 1962 Taiwanese and 1960 US U-2 shoot downs shows this was not the case for long, though it was still at the extreme limit of many existing systems and many less well equipped states simply didn't have the ability to hit the plane at altitude. Though the discovery of SA-2 batteries in Cuba in mid 1962 placed them in the former category and changed the calculus about flights over the island.
As for Cuba specifically the overflights were part of ongoing efforts to gather the latest accurate intelligence on the state of the Cuban military, and Soviet aid to it. Since before the Bay of Pigs invasion that had failed in April 1961 the US govt operated one plan or another for the overthrow of the Castro government, more broadly codenamed Operation MONGOOSE overseen at the very highest levels of the Kennedy Administration and more specifically by Attorney General Robert Kennedy. These ranged from supporting more expats landing on the island, a homegrown uprising, targeted air strikes, false flag jobs, or a larger invasion out of Guantanamo Bay or in response to any imminent threats to the base or Florida. While Soviet military aid to the island was public knowledge and it was desirable to monitor the extent and displacement of this aid. These overflights first spotted new air defense systems and new deliveries of modern jets like the MiG-21 during the middle of 1962. While human intelligence and other sources also supported the idea that a major Soviet buildup was on underway on the island and could include ballistic missiles which would be the game changing move that of course made the crisis what it was.
However following the ROC's loss of an aircraft that summer it was decided to pause overflights by U-2's from early September to mid October as it was clear that powers other than the USSR had now gotten or developed SAM's that could hit the jet, again the main system in question here is the SA-2 and its derivatives. And satellite photos and other methods including overflights of Soviet cargo ships proved unsatisfactory, while more signs that preparation for ballistic missile installations were happening built. Finally in mid October of 1962 flights were reauthorized to get better intel on what was happening and the first flight on the 14th with the film developed and identification made the next day confirming the worst.
I would also suggest some additional relatively easy reading for more on the thought process behind needing the routine flights over Cuba, stopping them in September, and then the debate and eventual approval of new flights. Both from the CIA one from 1963 relatively soon after the Crisis, and one much more modern. Both are pretty frank about competing agenda's limits of intel gathering methods, and political realities.
Modern history from the CIA about their and the U-2's role in the crisis.
Certainly not unbiased we must say but well worth the read.