Hint: the space program was mostly an effort to build reliable and accurate missiles that could be used to deliver the nuclear weapons anywhere in the world. The same technology that can launch an ICBM can also carry a couple of people into low Earth orbit when you put them on top instead of a nuclear warhead. So it makes perfect sense that the nuclear powers of the time would be spending so much time and money on developing a "space race" that was truly just about making accurate military missile technology.