What was the purpose of invention the jet plane?

It was simply a natural progression in aircraft power plant design and development. The British actually developed the first jet engine before WWII, but the Germans refined it even further and actually had several squadrons of jet equipped fighters by the end of the war. Their shortcoming was that they did not have much range due to fuel consumption and were easily shot down, when they were forced to try to land at their fields, by conventional piston-engine fighters who had a much longer "loitering" time in the air.