How did the kennedy administration know that the soviet union was putting ballistic missiles in cuba?

1) ariel spying over Cuba produced pictures that showed missile silos being built in Cuba. The design of the silos made it clear they were designed for missiles, and it made no sense for Cuba to put in anything less than nuclear missiles there. Missiles they could not build themselves, so had to come from the Soviet Union.
2) Only minutes. A launch from the Soviet Union to the US only takes about 20 minutes. Depending on the range of the missiles put into the silos, warning time would have been anywhere from 3-10 minutes. Not enough time to verify that it was a launch, and not a detection system malfunction, forcing America to launch immediately, or risk losing its capacity to strike back.
3) A direct attack or invasion of Cuba would have forced the Soviet Union to respond in kind. The USSR simply could not abandon Cuba, without losing all credibility among its allies and vassal states. So they would likely have struck back at the US, probably in Europe. This would have dangerously escalated the tensions, and increased the probability of nuclear war. Other officials believed that a quick,determined strike would not only eliminate the immediate threat of missiles in Cuba, but possibly overthrow the regime and force the USSR to accept the situation. The idea of a naval blockade was a compromise position. A threat of force, but one that allowed the USSR to back off. After all, so long as the missiles were not put into the silos, they were no threat.