Question

How did the cold war lead to the space race between the US and the USSR?

Answer 1

I believe it's the other way around with the Space Race affecting the Cold War. The launch by the soviet union of the earth's first artificial satellite sputnik 1 in 1957 into earth orbit shocked the united states 

Answer 2

The cold war lead to the space race by pitting each country against each other when they realized the military capabilities of objects in space. They could travel farther, faster.  They could have satellites loaded with nukes. Etc. So each country wanted to be the first up stuff. There was also that moral point for civilians because they each wanted to show their tech was better with the end point being to be the first on the moon.