What did Leo Szilard need to do in order to invent the atomic bomb?
As he did not have to build the atomic bomb, all that he had to do to invent the atomic bomb was to think.
That is really all that is required for most inventions: think of a new and novel way to do something.
What Leo Szilard invented that was new and novel was the neutron chain reaction, the process that in the atomic bomb drives the exponentially accelerating nuclear fission of uranium or plutonium atoms. However he did not know about either nuclear fission or what elements might be able to support the neutron chain reaction process that he invented. This took the work of others before the atomic bomb could actually be built.