Following the development of rocketry in the 1930s and 1940s, and its eventual deployment in World War II in the form of the Nazis' infamous V2 rockets, it became abundantly clear to the rising world powers of the time that rockets were the next big thing. Rockets are not the same as jets, which require oxygen in order to burn fuel and are thus limited to the atmosphere. Instead, a rocket can take up its own special kind of fuel that contains the oxygen it needs for combustion, and thus burn an engine out in the void of space.
Once this was possible, naturally humanity began looking skyward, ready to colonize the next frontier. However, as the Cold War rivalry between the United States of America and the Union of Soviet Socialist Republics kicked off, people began to wonder whether it might be a good idea to hurry up and get into space before their ideological opponents did. Partly, this was a matter of military security, since space is the ultimate high ground in any future war. Partly, it was about establishing one's own way of life as the right way and carrying the right ideals - democracy, liberty, free speech and so on - into the great wide universe.
People got themselves into space, and then into orbit around the planet. With the moon being the closest heavenly body, it became the next target: the first nation to set foot on the moon would be able to claim victory in the race. How well do you remember this giant leap for mankind?