First, the fossil record demonstrates that. Africa contained the first human fossils. Human fossils from other eras have also been discovered nearby but outside of Africa. Human fossils from later times have also been discovered outside of Africa. They were found in North and South America, and they date from relatively recent geological eras.
Second, because DNA evidence supports the same conclusion. There is a vast variety of genetic variation in Africa. Humans outside of Africa are all closely connected. The logical conclusion is that humans originated in Africa, and a small number of them—with low genetic variety due to their small number—left the continent.
A interesting tidbit about the exodus from Africa is that they interbred with other human species that did develop outside of Africa. Neanderthals, Denisovians, "hobbits," and maybe a fourth unidentified Homo species did evolve outside of Africa from pre-human creatures, most likely Homo heidelbergensis or H. erectus (the exact species is still up for question), which departed Africa and migrated over Europe and Asia. These non-sapiens Homo species developed from H. erectus, which, let's say, had left Africa; sapiens, on the other hand, descended from the same species, which had remained in Africa. The H. sapiens that left Africa did interbreed at a low level with Neanderthals and Denisovians and perhaps another species, despite the fact that they were distinct species and as such were very poorly interfertile with each other. As a result, non-African peoples, but not African peoples, have a small amount of non-human ancestry.
Small morphological and genetic changes found in the largely inbred populations that emigrated from Africa are readily explained by natural selection, sexual selection, and neutral or nearly neutral drift. In contrast to the far greater genetic and morphological diversity seen within Africa, I mean by this that the period of time that these people have been known to have lived outside of Africa is easily long enough for natural selection, sexual selection, and genetic drift to bring these changes.
None of this is controversial any more (except for the fine details of exact timing, exact patterns of migration, and so on); it's basic, introductory material that has been examined and confirmed in many different ways.