Hey there!
To understand the Space Race, you must first understand the Cold War.
The Cold War wasn't really a War - in some views, it was just a competition. The US and the USSR were the largest superpowers the world had ever known - they had the most weapons, money, and power, ever. The US and the USSR were engaged in an arms race initially and were each racing to get the most nuclear weapons.
That was until the Soviet Union launched the Sputnik. The US could not let the USSR get control of space, and in 1969, launched the Apollo 11 to combat the USSR who had recently put the first man in space - by putting the first man on the Moon.
In my opinion, even though it was a competition, it was worth it. Think about it- would we have sent a man to the Moon afterward without all that pressure? Even nowadays, we would not have discovered what we already discovered without it and would not have sent any more rockets. Despite the cost, it has benefitted and will continue to benefit us in the long run.
Hope this helps!
