25,000 Cuban army,200,000 Cuban Militia, 9,000 armed police against 1500 CIA sponsored counter revolutionaries and 8 B-26 bombers.
I understand that it was a failure, but how and why would the US approve such an attack (damaging their foreign relations) against these odds?
To clarify: This is not a question about US anti communist policy but why would the US engineer such an attack and expect any other result, It seems more like a token effort than any real commitment to overthrowing Cuban communism.
Much in the way that Castro had succeeded against similar odds years earlier. The Bay of Pigs was not the original target of the attack, but rather, Trinidad, Cuba, which is on the southern shore roughly in the middle of Cuba. The crucial geographical difference between the two is that the Trinidad Plan allowed for the rebel forces to escape into the Escambray Mountains, which had previously been used to great effect in conducting guerrilla campaigns, while the Bay of Pigs Plan allowed for the air support to be done without air refueling and thus keeping the US footprint lighter. Once the guerrilla fighters planted themselves on Cuban soil and secured a stable area, they were to create a temporary government and petition to the United States for assistance.
President Eisenhower originally had the CIA develop the plan and begin training the rebels, but the decision came to Kennedy whether to carry it out. This is where we get to your question - why did the US go through with the attack when the odds were not looking so great? The answer is political. When Kennedy was campaigning against Nixon, he accused Eisenhower of standing by during the Cuban revolution. He could not afford, politically, to scrap this plan, after making that an issue.
This is where we get into WHY the attack failed, or at least, what didn't help it. Kennedy wanted the footprint light. He and his administration ultimately rejected the Trinidad landing spot and went with the Bay of Pigs, but they also reduced the air sorties from 40 to 8. The Secretary of State, Dean Rusk, and others in the administration and military advisers began to think this was an ill advised plan, especially since Castro was proving to be quite popular. There was no way this invasion would succeed without US support, but Kennedy insisted that US deniability be made the top priority.
In short: because Kennedy made a choice and his administration and advisers did not voice their opposition to the plan, at least not in a significant manner.
Source: Combs, Jerald. The History of American Foreign Policy from 1895
Edit: I forgot a word in the book title... FOREIGN policy. ooops.