Homo sapiens left Africa more than 100,000 years ago, but they left hardly any genetic traces behind. The eruption of the Toba volcano in Sumatra caused the climate in the eastern Sahel region - and probably in other parts of Africa - to become significantly drier.
People could have migrated out of Africa along the Nile and north of the Red Sea, but also via the Bab-al-Mandab Strait on the Horn of Africa. This strait is currently around 26 kilometers wide and may have been significantly narrower or even completely dry back then.