Well, "history" is the study of text, but generally that means that it only goes back as far as there has been written word (and myths and oral histories etc). In Western Eurasia, this goes back to the Sumerians in about the fourth millennium, although one could argue that you still can't really do "history" for some time afterwards. However, we can extend our knowledge of past societies far back before that with archaeology, the study of past material remains. Archaeology can tell us a great deal about the societies it studies, such as economies, political structures, and religions.

Now, to answer your question, the period you are curious about is called the "Neolithic" which is a word essentially meaning "new stone age" and is a period of time after the development of agriculture and before the use of metal. Çatalhöyük is southern Turkey is probably the most famous Neolithic site in the "Fertile Crescent", but an important thing to remember is that there is no "one thing" happening at this time. You are asking about the entire surface of the earth, a whole bunch of things were happening, and the spread of agriculture around the Black Sea had about as much to do with what as going on in Florida as the Lapita settlement of Tonga had to do with Homer.

As a side note, please disregard the Ancient Aliens nonsense in the other response.