As you can see from the link below, nothing happened on October 5, 1957 that is particularly noteworthy in the history of the world. I wonder, however, if you might be thinking about the previous day, October 4, 1957. That date is very important in world history as it was the date on which the Soviet Union launched Sputnik, the first artificial satellite to orbit the Earth.
The launch of Sputnik was important because it was a scientific/technological breakthrough, but it was also important for geopolitical reasons. At this time, the US and the USSR were locked in the Cold War. Each side wanted to outdo the other and each side wanted to gain a military advantage over the other. The launch of Sputnik was important in this context. It made the US look bad and the USSR look good because the latter had been first to manage to launch a satellite. It was also militarily important because the launch showed that the Soviets had the ability to build rockets that could be used to carry nuclear warheads. For these reasons, the launch of Sputnik was very important to Americans of the time.
Because Sputnik was such an important event, and because there is nothing really important that happened on October 5, 1957, I wonder if the launch of Sputnik is the answer for which you are looking.