JFK did not broadcast this statement in a speech, but rather, he made the statement in an interview with a journalist, one-on-one.
The statement was in regards to Vietnam, shortly after the [C_A-paid] Viet generals took, and killed President Diem. First, he said that in the final analysis the Vietnamese people would have to be responsible for their country, and it's government. Then, having already been double-crossed by C_A director A. Dulles in BayOfPigs fiasco, JFK made the statement in regards to what he would have to do before he, JFK, could continue the build-up of US armed forces in South Vietnam.
We had about 25,000 military (&C_A) 'advisors' .. three weeks later JFK went to DALLAS.