The Short Version
The claim is well-supported by extensive scientific evidence. During recurring "African Humid Periods" — most notably roughly 11,000–5,000 years ago — large parts of the Sahara had significantly more rainfall, flowing rivers, lakes, and water-dependent wildlife including hippos, crocodiles, and large game. However, the phrasing slightly overgeneralizes: these green conditions were episodic rather than permanent, and geographically uneven rather than uniform across the entire desert.