Why did the US become increasingly involved in Vietman?

Help me PLEASE!!!

Japan had a controlling interest in Vietnam up to 1945. After they were defeated in World War Two they no longer had control over Vietnam. Part of allies already had involvement/ interest in Indo-China, so they planed to take over Vietnam.
After 1945, the U.S was already dominated by their fear of communism.
The Americans came up with the idea, which was to be referred to as the domino theory. They thought that if Vietnam fell into communism then Laos, Cambodia, Thailand and possible even India would fall into communism (just like a row of dominos). They were determined to resist the spread of communism in Vietnam because this was seen to be the first domino in the row. Their policy was containment, the most important of all being the Truman Doctrine. President Truman promised the U.S would support any country with money and arms that were fighting communism. Their methods and policies showed their ignorance of the Vietnamese people and the region.
France was one of the countries. They were fighting communism because when they occupied Vietnam, Ho Chi Minh, a communist leader, formed a rebel group called the National Liberation Front, to fight the French and free Vietnam from their control.
In 1955 they helped Ngo Dinh Diem set up the republic of South Vietnam. The votes were obviously rigged because only 450,000 people were eligible, but Diem managed to get 605,000 votes! This was because he was anti communist and was prepared to imprison or kill communists. He didn’t show any respect for the Vietnamese peoples beliefs and religions.

I'm doing an essay for history but this is as far as I got. I showed my teacher this and she gave me some brief not but I don't understand them. could somebody please try and explain them to me:

*V.C- civil interest/open rebellion by vietnamese peasants

*Kennedy sent advisors after death of Diem (1963)

*Assasination of Kennedy let to attitude by president Johnson (told the US navy to patrol coast of north vietnam to make their presence felt)

*Patrols along N>Vietnamese coast

*Gulf of tonkin

*Tonkin resolution

*Rolling thunder- direct troop involvement.