JFK ascended to the presidency in 1961. Just like many other presidents, he was in turn given many conflicts which earlier presidents had gotten themselves involved in and in this case, the American commitment to Vietnam had already been active for 11 years. Starting in 1950, the American commitment had first involved spending large amounts of money to support (as well as supplying equipment) to the French Expeditionary Force in French Indochina. When the war came to an end in 1954 and the French left what became North Vietnam, South Vietnam, Cambodia and Laos - the United States decided to take over France's role in the newly formed South Vietnam and to protect this ally of the so-called Free World from a possible invasion from the North. The first military advisors sent to South Vietnam were sent by the Eisenhower administration and the JFK administration essentially continued an already set pattern.
The reason for the increase in both advisors, special forces and non-combat personnel was simple: the reality on the ground had changed. Since the late 1950's, insurgents had been targeting both civilians and military alike. What had begun as local assassinations against government officials on the country side escalated over the years and the insurgents also began to attack regular ARVN (Army of the Republic of Vietnam) units. Just a month before JFK took office, the formation of the FNL (Front National pour la Libération du Sud Viêt Nam - or more commonly known as the Viet Cong) took place. As the insurgents in the countryside and in the cities grew in strength, so did JFK's commitment to the ARVN's fight in combating them.
The political reasons were just the same as those set out by Harry Truman in 1950 and Dwight D. Eisenhower in 1955: To contain communism, to prevent it from spreading to the rest of Indochina and to live up to the image of the most reliable ally in the fight against communism. If South Vietnam fell because the US did nothing, it would have seriously tarnished their image of a protector of democracy and freedom.