Jared Diamond's "Guns, Germs and Steel" is attempt to explain some of the apparent differences in the development of various human cultures; why were some cultures sailing the open oceans and developing gunpowder while others were still hunter-gatherers using stone tools?
It should be noted that Diamond has received significant criticism for some of his opinions, and the picture of human evolution is a constantly-shifting work in progress. For an idea of the professional arguments leveled against Diamond, see the link in my sources below, "Why Does Jared Diamond Make Anthropologists So Mad?".
To answer the question, "why did humans emigrate out of Africa?", we should first clarify what we mean by "human". There were many human ancestors and relatives who migrated out of Africa. Modern humans, homo sapiens, are currently thought to have originated in Africa, and then migrated outward, displacing their relatives in the process. To our knowledge, homo sapiens achieved the broadest migration of all human relatives, and was the only one to reach Australia and the Americas. Diamond attributes this success to the "Great Leap Forward" - a sudden surge in apparent "humanity" as exhibited by trash heaps that bear distinctly "human" artefacts.
Curiously (and perhaps, this is part of the reason for criticism of Diamond) he does not actually state why humans left Africa!
Migration is not an unusual thing, though; there is extensive evidence of migration in the fossil record. For example, it is thought that horses and camels originally evolved in North America, migrated, and then went extinct in North America while surviving in the places that they had migrated to. Common reasons for migration include escaping poor weather, seeking more abundant food or a less competetive environment, or natural disasters that cut off a population from a particular area.
In the case of human evolution, it is thought (again, with caution against making too many assumptions) that the Great Rift Valley may have provided a sort of ecological cradle for humans, and connected to a belt of grassland ecosystems reaching as far as Asia. There may not have necessarily been any intent to migrate; remember that humans didn't have a concept of "Africa" at this time period, and they may have simply been looking for new resources and new living space, or just satisfying their developing sense of curiosity.