Question

how did the kennedy administration know that the soviet union was putting ballistic missiles into cuba

Answer 1

The United States had aerial photographs taken by Major Richard Heyser, flying a U-2 plan on a surveillance mission.  

The U-2 aircraft was made by Lockheed for flying at extremely high altitudes, developed to fly out of range of enemy radar or anti-aircraft weapons.  The first U-2 flights had occurred over communist territory in Europe in 1956.  

After the photographs by Major Heyser were reviewed in Washington in October, 1962, there was intense concern in the US government.  A second U-2 mission was sent, and additional intelligence was gathered about the type of missiles the Soviets were locating in Cuba.

The Cuban Missile Crisis had begun.