I imagine they had tens of thousands of soldiers and corresponding baggage train. Many areas of North Africa andthe Arabaian peninsula and the middle East are just deserts or scrubland with not much water or vegetation. Not a ton of farmland in the homeland either to take grain on the March with them.
Same way as the large Army question you've already answered. Yes there are vast swaths of uninhabited land but conquest is primarly overtaking cities and kingdoms which lie in the cities.
Oasises and Rivers are the key to building up from town to city to Kingdom. Following the trade routes an Army can exploit these resources along the way.