The Pentagon brass felt humiliated by the failure of the Bay of Pigs invasion... they felt they had much to prove and it could only be done by the removal of Fidel Castro from Cuba.  Operation Northwoods was a plan drawn up by General Lyman Lemnitzer and signed onto by all the military leadership to create one or more false flag events that could be used to justify military action against Castro's Cuba.

Plans ranged from killing some Cubans in Miami to blowing up a boat full of Cuban refugees to taking down a commercial airliner and killing all Americans on board... or at least making it look like that happened. The point was to generate public support for removing Castro and his terrorist government. Every member of the Joint Chiefs of Staff signed off on the document that was presented to President Kennedy.

The president's response was an emphatic refusal to invade Cuba... and, more importantly, we are not going to kill Americans to justify it. This was just one more step in the division between the Commander-in-Chief and the Department of Defense as Lemnitzer was banished to run NATO in Europe.

A case study of Operation Northwoods and Operation Mongooses and its ethical implications
by Jonathan White

This case study is an in depth analysis of two classified operations that took place or almost took place during the Cold War which posed very difficult ethical challenges for the United States. Even in the darkest parts of American History we can, as professionals, extract lessons learned to apply in future conflicts.

Operation Northwoods: The History of the Controversial Government Plan to Stage False Flag Attacks on Americans and Blame Cuba
by Charles River

The severity of false flag operations moves in tandem with the perceived danger. In the late 20th century, the Soviet Union constituted an ongoing threat to the United States, ideologically and militarily. When Fidel Castro, a Soviet sympathizer, came to power in Cuba in 1959, the presence of a communist extension of Soviet influence so near American shores caused great discomfort to the U.S. government.

Some of the Cold War’s tensest moments involved Cuba, including the Bay of Pigs invasion and the Cuban Missile Crisis, amply demonstrating that throughout the early 1960s, alarmed individuals within the American government believed that the only way to handle Castro was by immediate elimination and a takeover of Cuba. They were pitted against more nuanced points of view held by others within the Kennedy administration. The Pentagon was of the general mind that in such a perilous environment, military decision-making should come to the front and civilians should withdraw, even the White House. Diplomacy was seen as being of little use by the Joint Chiefs of Staff against an encroaching Soviet superpower led by a political street fighter and his puppet situated just off the coast of Florida.