The United States interfered with the Korean War because of 3 reasons.
1. North Korea acted as a buffer zone between communist china and japan.
2. Due to the Domino Policy they believed if all of Korea fell to communism the rest of asia would too and that needs to be stopped.
3. It wanted to show the world that it was still a very strong country and wanted to show russia and china that it will stop the spread of communism and it will not allow it to spread.