As far as I can tell...

The Who:

According to Jim Garrison, there were three principal groups. Anti-Communist Cuban Exiles, EXTREME Right Wing Extremists, and former or unofficially acting CIA Operatives. He also had reason to believe members of the Mafia had a secondary role. In addition, there are journals that implicate certain wealthy Texas Oil Tycoons.

The Why:

JFK didn't send in air support during the Bay of Pigs, which turned out to be a disaster. But apparantly the CIA was never authorized to plan for its use. At the last minute they seem to have told JFK their plans required it. Kennedy wouldn't give in to the demand. The Bay of Pigs and other related Anti-Cuban activities involved the hiring of the Cuban Exiles, Right Wingers, and, to a lesser extent, even some Mafia related individuals. This is in addition to the CIA Operatives who organized the Anti-Cuban group. The CIA was notorious for finding alternative funding for its operations. This may have come in the form of Oil Tycoons who wanted to aid their nation.

So first, a substantial portion of this paramilitary group was killed in the Bay of Pigs. Then Kennedy began cutting funding to the anti-Cuba/Castro operations, and finally ordered their complete halt. His call for a "cooling of hostilities" with the Communists was a betrayal of the original promise made to the paramilitary enlistees. Startlingly, the New Orleans contingent specifically specialized in assassination techniques. They lashed out at the leadership of the operation. This sort of phenomenon is commonly referred to as a "Blow Back Effect" and is a common problem in intelligence work. President Kennedy was the primary target since he was supposed to be the head of the Anti-Cuban activities, even though Bobby was actually the unofficial liaison between the Whitehouse and the CIA.

The FBI had been tipped off on the assassination but hadn't acted, so they had to cover it up. Bobby had been the middle man between John and the CIA, so he had messed up and wouldn't let it out. The CIA was obviously incompetent, so they had to cover it up. And LBJ had become the politician that he was because of Texas Oil Tycoons' money, so he couldn't let it out, either.

Thus, you have the assassination and the resulting cover up -- Blow Back Effect followed by the ones in charge covering their neglegent butts, respectively.