How did the war between japan and russia result in the cold war. how was the us and china involved exactly?

The war was actually between Japan and USSR, not Russia. It was fought on land belonged to China with US and Western allies watching and supporting the Chinese govt.

Japan was the aggressor and invaded part of China bordering USSR. To protect its own interests in China, USSR fought off Japan and took over the puppet govt in Manchuria set up by Japan. It allowed USSR to spread communism to China which turned into a communist country later. US and Western allies won WWII but were alarmed to see the spreading of communism. That confrontation was the beginning of the cold war.