The specific incident usually pointed to as the start of U.S. involvement in the Vietnam war was the Gulf of Tonkin incident. A pair of American destroyers stationed in that gulf reported via radio that North Vietnamese military forces fired at them. Lyndon B. Johnson was president at the time, and used these radio reports to ask for Congressional approval to ramp up military presence in the area. This all occured in August of 1964, and paved the legal pathway that both Johnson and later Richard Nixon used for military action in Vietnam.
The United States had a number of motivating factors to get involved in Vietnam's internal conflict. China had succumbed to communism shortly after WWII ended. There was concern in Washington that if Vietnam followed suit, that communism would continue to spread throughout southeast Asia, possibly even into India.