Well, firstly, you have to consider that the environment and climate of the Middle East and West Africa thousands of years ago was not exactly identical to the climate of today. Sure, it was still very hot and arid in places, but it was definitely habitable. That's why some of the first states in history formed there.

While I could give a very long, essay answer as to how people in the Middle East and West Africa (my knowledge rooted primarily in the former) lived, needless to say it rested a lot on the major rivers. Primarily, for the Middle East, the Nile, Euphrates and the Tigris. These provided fertile soil for people to establish farms and grow crops, and in fact the basin known as the 'Fertile Crescent' between the Euphrates and the Tigris was a great area for animal grazing. All of this (and much more) allowed for the establishment of an agriculturally-based system and economy which people living in the area would live off successfully for thousands of years.

I can't say much for West Africa, but that is a very basic overview of how it worked in the Middle East / Ancient Near East.