I'm basing my answer on both having grown up during the Cold War and marrying a woman who grew up during the Soviet Union.
Right after WWII, even though there was a brief respite between the U.S. and their allies, the Soviet Union, the rivalry of the Cold War soon appeared.
When North Korea attacked South Korea, a communist/dictatorial country attacked a Democracy. The Soviets were elated. One less Democracy and a defeat for United States of America who propagated Democracy.
While not militarily involved, the Soviets sold all necessary equipment to North Korea to help them achieve victory.
Americans knew this would be a major loss and a huge boost for communist propaganda.
As Americans launched an offensive to retake South Korea, the Soviets were angry since their rationale was that Americans were putting their nose into others affairs.
Also, in October 1950, Communist Chinese troops under the name of the Chinese People’s Volunteer Army (CPV) crossed the Yalu River to assist North Korean armies, and engaged in the Korean War in an offensive manner after the U.S. troops crossed the 38th parallel. This also contributed to the beginning of China's role as a nemesis in the newly coined Cold War.