Why was Sputnik so important for the Soviet Union?

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.