How did the Korean War start?
There are a number of possible ways to answer this.
The answer that you are most likely being asked for is that the Korean War started when North Korea invaded South Korea. This was the immediate cause of the war. On June 25, 1950, North Korean troops attacked across the border between the two countries. The South was very poorly prepared and the North advanced rapidly through the country. This was the start of the Korean War.
However, it is also possible to say that the war really started earlier. IF we say this, we are looking not at immediate causes of the war but at its underlying causes. We could say that the underlying cause of the war came into existence when the Soviet Union joined the war against Japan in August of 1945. The Soviets were not involved in the war for very long, but they were involved long enough that it was their troops that occupied the northern part of Korea when the Japanese surrendered. This led to a situation in which Korea was split into two countries, one ruled by communists and supported by the Soviets, and one non-communist government supported (though not very strongly before the war broke out) by the US.
That brings us to one other possible answer. We could argue that the war was brought about by US disinterest. The US pulled its troops out of South Korea in the summer of 1949. The US Secretary of State also gave a speech in January of 1950 that implied that the US would not defend South Korea. This lack of interest on the part of the US can be seen as a cause of the war as well.