Not sure if this is the right sub to ask this, but I just got to thinking of this. If early humans were adapted to African climates, why did they migrate/spread to places with much different climates, such as Europe, Asia, present-day Russia/Siberia, etc.? Was it a matter of searching for better food sources (following herds, food shortages in their original region of Africa)? Competition between other humans? Or did early humans just sort of expand their range until they unintentionally wound up in such places? Again, I'm not sure if this is quite the right place to post this sort of history question, but I couldn't find a better one.
You may have better luck with /r/AskAnthropology. I'd recommend asking there as well.
Not really a history question, but I'll give it a shot.
During that period of time around 100,000 years ago, there were drastic climatic changes occurring. Africa was drier than it had been before, leading to much of of the landscape changing. Sea levels were lower at that point, meaning what is now the Red Sea was more like marshland.
As humans were hunter-gatherers at that point, they followed the herds that they hunted, like you mentioned meaning that as the animals moved out, maybe from the climate change, so did the group that hunted them.
From what we can reconstruct from genetic information, it seems that a relatively small group of individuals, possibly around 1,000-5,000 left Africa and went to the Middle East, where they spent quite a long time (and got it on with some Neanderthals). Then they moved down along the coast to India, South Asia, and Australia, where at some point they met and got it on with some Denisovan peoples.
Following that there were migrations to the rest of Asia once the ice covering it thawed and finally to Europe around 40,000 years ago.
This isn't a history question. These events occurred a LONG time before history. Anthropologists study all of humanity's past; historians only study history.