Why was the fair so important, especially given the time, to both Chicago and the United States?
There are a few different ways to answer this.
For the city of Chicago, the fair was very important economically. The United States was in something of a deep recession at the time. Because of that, the visitors who flocked to the fair from other parts of the country were very welcome. They put a lot of money into Chicago's economy.
For the US, the fair was something of a declaration of power. It emphasized American superiority and world prominence at a time when the US was trying hard to become an imperial power and an important player on the world stage.
That fair, as with all world's fairs and the Olympics later, would bring renewal to Chicago - new construction, new jobs, new money coming in to the city more tourism, more publicity. In addition, fairs at that time were a place to showcase new inventions and technologies, which was a great way to also increase world trade and open up new markets.
Over the long run, it did give us a slice of the world stage at a time our empire was being born, and served notice to the world that America's ingenuity, economy and military were to be taken seriously.