The most important point to understand about the “space race hoax” is that it does not exist. There is no such thing as a space race hoax. The term “space race hoax” can be used to refer to the idea that, at the very least, the US moon landing in 1969 never happened. This is, of course, simply untrue. American astronauts did land on the moon in July of 1969 and then did so five more times between then and 1972. Since people really have been on the moon, there is no hoax.
The other important point to understand, then, has to do with why people think that the space race and/or the moon landing was a hoax. In general, conspiracy theories like this come about because people do not trust their government and because they are made uneasy by the complexity of society. In large, impersonal societies like our own, conspiracy theories are bound to spring up as people feel a lack of connection to their government and to others in their society. This is why we have conspiracy theories about such things as the Kennedy assassination, the 9/11 attacks, and the shooting at Sandy Hook Elementary School.
So, the most important points about the “space race hoax” are that A) there is no such hoax and B) people believe in that and other conspiracy theories because of their distrust of government and unease with the complexity of modern society.