What became the most important adaptation for hominins that moved out of the african forest onto the savannah?

You might be talking about the Out-of-Africa Theory or early migration of early modern human species out of Africa. According to this theory, our modern human ancestors left Africa about 200,000 years ago. They left in two waves; one died out or retreated while the second populated the world. At that time, Africa experienced very long droughts, so early humans have to migrate to other greener areas. Therefore, Africa at that time is not forested and humans did not move out onto savannah regions.