How did the space race help to ease Cold War tensions?
This is not a position that is commonly taken by historians. Instead, it is more common for historians to argue that the space race increased tensions by making each side worry that the other side was gaining an advantage. This was important because rocket technology for the space race could be (and was) used to make better delivery systems for nuclear weapons.
There are at least two ways to argue the opposite, though. You can say that competition to get into space diverted the two sides. It gave them a way to compete for prestige instead of competing for power, which would be more likely to lead to war. Second, you can argue that satellites from the space race, like the CORONA reconnaissance satellite, allowed the US to determine what kinds of weapons the Soviets had and in what numbers. This eased tensions by decreasing US worries that the Soviets were gaining an upper hand.