Sputnik was important for the Soviet Union for two main reasons.
First of all, Sputnik was somewhat important in military terms. By creating a rocket that could reach space, the Soviet Union was moving ahead toward having missiles that could carry its nuclear weapons. This was more important for the Soviets than for the US because US nuclear weapons were lighter and could be carried by bombers. By contrast, Soviet weapons needed missiles to carry them. Therefore, Sputnik represented a major step in Soviet military development.
Second, and more importantly, the Soviet victory in the space race was a major public relations victory. By launching Sputnik, the Soviets were proving (they could argue) that they were equal to (or better than) the US in terms of science and technology. This gave them a great deal of prestige in the eyes of other nations.