The launching of Sputnik by the Soviet Union in 1957 is connected to the Cold War because it was an example of the competition between the US and the USSR during the Cold War.
During the Cold War, the US and the USSR competed in a variety of ways. They competed militarily, for example, and for prestige in the eyes of other countries. The launching of Sputnik was a win for the Soviet Union. The Soviets managed to be first to launch a satellite into space, which was seen as a triumph for Soviet society and technology. After the launch, the US tried hard to catch up to and pass the Soviet Union in terms of space technology.
The launch of Sputnik and the reaction that it caused are very typical of the Cold War competition between the US and the USSR.