How did the transition from hunter gatherer to stoneage man take place?

"Stone-age man" is a very broad term referring to any number of peoples who used stone tools beginning several million years ago in Ethiopia and appearing elsewhere from their. Most stone age men were not even homo sapiens, homo habilis meaning "handy man".

To answer your question then, stone age men and hunter gatherers are the same thing. The period following the stone age was the copper age, followed by the bronze age.

What you are probably wondering is how nomadic people like hunter-gatherers transitioned into agricultural settlements. Very early forms of farming are found in the Indus and Nile river valleys, and a couple of other rivers in China that I can't remember right now. Eventually people in these very fertile areas began realizing that if they lived where their plant sustenance cam from, they would no longer have to move all the time to follow their food source. Further more, they found that herd animals could be easily subjugated and kept as livestock, thereby fulfilling their needs.