What were the Soviets doing in Cuba and how did President Kennedy find out?

The Soviets were essentially constructing nuclear missile sites in Cuba. President Kennedy discovered the clandestine Soviet activity after an American U-2 spy plane managed to get photographic evidence of a Soviet SS-4 medium-range ballistic missile being assembled.
The spy plane obtained the devastating information on October 14, 1962. President Kennedy was briefed about the discovery two days later on October 16. Then, on October 22, the American public was made aware of the crisis at hand.
At the time, the Soviets were closely allied with the Castros in Cuba. To the Soviets, Cuba was a strategic location. However, to the Americans, nuclear-armed Soviet missiles launched from Cuba were a frightening prospect; the missiles could target most of the eastern United States. For their part, the Soviets looked at Cuba as an insurance policy against the American missiles targeting the Soviet Union (from the vantage point of Turkey and Western Europe).
In the end, Soviet leader Nikita Khrushchev (1894–1971) reached an agreement with President Kennedy. The Soviets would remove their missiles from Cuba provided that the Americans would promise not to invade Cuba. Privately, President Kennedy would also agree to remove American missiles from Turkey. Although the agreement led to a peaceful resolution to hostilities, the Cold War was far from finished. Even though the Soviets removed their missiles from Cuba, they later began investing in the development of intercontinental ballistic missiles that could travel from the Soviet Union to the United States.
https://www.history.com/topics/cold-war/cuban-missile-crisis

https://www.jfklibrary.org/learn/about-jfk/jfk-in-history/cuban-missile-crisis