Why did Enrico Fermi invent the nuclear reactor?

This was part of the Manhattan Project in WW2 to develop the atomic bomb. You must remember that at that time science was being pushed hard to come up with answers in order to understand basic nuclear facts, with the aim of making a bomb. Nobody even knew if it was possible, but the US was afraid that Nazi Germany might get there first, so every avenue was looked at for answers. It was known that enriched uranium, in fact nearly pure U235 would be needed, and this was done by using the gaseous diffusion process with uranium hexafluoride. The other route, only known theoretically, would be to produce plutonium 239 by irradiating uranium 238 in a neutron flux. The way to get enough neutron flux was to produce a continuous controlled neutron chain reaction, and Fermi was the scientist who had the best ideas how to do this. The first 'pile' as it was called was the result, built in Chicago as an experiment to prove the theory. This used a graphite moderator and metallic natural uranium fuel. It worked as predicted, but to get enough plutonium something much bigger and more powerful was needed. The same type of reactors but more powerful were therefore built at Hanford in Washington state and these produced the plutonium used in the second atomic bomb (Nagasaki) The following para is taken from Wikipedia article on Manhattan Project:

The first major scientific hurdle of the project was solved on December 2, 1942, beneath the bleachers of Stagg Field at the University of Chicago, where a team led by Enrico Fermi, for whom Fermilab is named, initiated the first artificial self sustaining nuclear chain reaction in an experimental nuclear reactor named Chicago Pile-1. A coded phone call from Compton saying, "The Italian navigator [referring to Fermi] has landed in the new world, the natives are friendly" to Conant in Washington, D.C., brought news of the experiment's success.