How did the US help South Korea?

They kept the country from falling into Russian style Communism.

The US during WWII enabled Korea to get free from Japanese colonialism.
During the Korean War, the UN countries and the US sent aid and soldiers, and with help from numerous but ineffectual South Korean soldiers kept Russia/China aided North Korea from capturing South Korea.

People at the time (1950-53) weren't sure at the time that this was necessarily a good thing, most modern Koreans are thankful that South Korea did not fall then. The US sent the most soldiers and aid, and suffered the most losses.

Korea has unrepayable debt to the US and to various UN nations for the aid and lives they provided in the Korean War. Every Korean feels this deep down, though the expression during various protests may not seem that Koreans remember.

-opinion-
However, the US is accepted as a big brother to Korea, and will always be a loyal sibling though they get obnoxious at times.