Why was NASA developed?

NASA was created in 1958 as an outgrowth of the National Advisory Committee for Aeronautics, and primarily as a direct result of the Soviet Union's lead in developing technology to send humans into space, in particular the launch of Sputnik 1.

After the successful Soviet launch of the Sputnik 1 satellite in 1957, the Eisenhower Administration formed the NACA to investigate what a new federal agency dedicated to putting the U.S. in outer space should entail. The result of the committee's recommendations was NASA.

It should be noted that it wasn't just space exploration that was being focused on at the time; the U.S. knew that any county capable of sending anything into space would also have nuclear ballistic missile capability, and staying ahead of the technology curve was paramount during the Cold War.