With the North Korean invasion.

That's the easy answer. It, of course, had a background and somewhat deeper origins. Tensions were basically inevitable when the peninsula was first partitioned between Soviet and US zones of occupation in the immediate aftermath of the Soviet invasion of Manchuria and the surrender of Japan. The North came under the rule of Kim Sung-Il, who was both a diehard communist and a Korean patriot who believed in only one Korea (under his rule). The South was ruled by Syngman Rhee, a rather autocratic figure and also a Korean patriot who believed in only one Korea (under his rule). Low level conflict, including both sides crossing their shared border under arms, began early. What we call the Korean War was only one phase in what was already a nearly five year old conflict by that point. Of the two sides, the North was well armed and equipped by the Soviet Union. At the same time, South Korea was basically starved of any major arms and materiel by the United States.

Many have pointed to Dean Acheson's speech delineating the extent of US security concerns as important to the immediate cause of the Korean War (as opposed to its deeper origins), as he did not mention Korea in that speech. Thus, the argument goes, the communists either believed him or decided to test whether this was true. As it occurred, the Americans did care, and so intervened to save South Korea. In all likelihood, the US government had given only a smidgen of thought at most to the question prior to the invasion--it was an academic question and so probably got only a minor academic answer. Once it became a real question of policy, it got a real answer. By 1950, of course, Truman and the Democrats had "lost" China to the communists under Mao, which led to major attacks on them by the Republicans and undoubtedly added further incentive to intervene when the North invaded the South.

There's also Stalin's role to consider. Many Americans in power at the time believed that it was an invasion orchestrated, not in Pyongyang, but in Moscow. Kim Sung-Il had been quite keen to invade the South for years, and had repeatedly requested permission from Moscow but was denied. By 1950, perhaps because of the communist victory in China and so with the concomitant assumption that China would help the North if necessary, Stalin finally withdrew his objections (although whether he actually endorsed the invasion I'm not sure has been really established, although the presence of some Soviet fighter pilots suggests that he at least eventually did).

Thus the North invaded the South. The Americans intervened soon afterward, and the Korean War as we know it began.