How do we know the human species arose in Africa?
There are two big prongs of the out of Africa theory or whichever name you wish to call it.

Prong the first: fossil evidence. There are lots of different kinds of protohuman fossils. Homo erectus/ergaster are found all across Europe, Asia, and into Indonesia from about 1.5 million years or so until about 70 thousand years ago, where they stop showing up. Up until 125 thousand years ago anatomically modern humans were only found in Africa. About 70 thousand years ago they start showing up in the Middle East, and then they spread out roughly following these paths. The fossil record shows not-humans or almost-humans (Homo erectus mostly) for millions of years, then humans show up at a certain point and then there's loads of human fossils. This point gets more recent as you get further from East Africa.

Prong the second: genetic diversity. Starting with this paper (not sure if this is behind a paywall but the details are not important anyway) human mitochondrial DNA was compared using restriction enzyme mapping. This kind of mapping is super crude (roughly analogous to grinding pottery into a fine dust and grouping the dusts by color) and the original study was pretty limited by the computers available at the time. Nonetheless, they showed that the most genetically diverse geographical group was Africa, and furthermore that any two people from outside Africa are likely more closely related to each other than any two people inside Africa. Genetic diversity goes down the further away from East Africa you get, matching what you'd expect if humans hadn't been living there long. There have only been (very roughly) 60 generations since humans colonized New Zealand. All Maori people, therefore, are at most 58th cousins. All indigenous Australians are at most 1998th cousins. All Africans are at most 8000th cousins.

More recently we've done proper sequencing on the mitochondrial DNA (actually knowing whats written there) and that's allowed much more precise estimation of this sort of thing, and estimation of the timeframes involved. The genetic evidence backs up the fossil record, by and large. (There are a few interesting examples where interbreeding with Neanderthals and maybe Denisovans can be detected, and that the settlement of the middle east by modern humans about 125 thousand years ago failed. They left no descendants.)

Fun side consideration (warning, extra science, proceed with caution): all humans inherit their mitochondria from their mother, and all males inherit their Y-chromosomes from their father. This lets us trace matrilineal lines with a relatively high degree of accuracy, since mitochondria reproduce asexually. To clarify: your matrilineal line is your mother, and her mother, and so on. Your paternal grandmother is not part of your matrilineal line, nor are your aunts or sisters. These lines are statistically guaranteed to eventually converge on a single woman whose mitochondrial descendants now live in all of us. The reasons for this are complicated. There are actually lots of these women, since she inherited her mitochondria from her mother, ad infinitum (well, not infinitum. Ad failed-endocytosis-of-an-alpha-proteobacterium-leading-to-mitochondria doesn't roll off the tongue though). Science has tight estimates on the most recent mitochondrial ancestor, or "Mitochondrial Eve". To be clear: there were loads of other women alive at the time, and we inherit a lot of their non-mitochondrial DNA. Over millennia all those other women had fewer daughters, and their matrilineal lines died out. She lived about 200 thousand years ago, plus minus 20 thousand years. As a mathematical guarantee we'll never find her body or have a really tight estimate on how long ago she lived, but she's genetically guaranteed to exist. The same principle applies to patrilineal inheritance, but the estimates for the most recent ancestor of all Y-chromosomes are a little looser (200-300 thousand years ago). See here for a more detailed handling of Eve and what's going on there.

In sum: old fossils only in Africa, everyone outside Africa is only 2800th cousins with each other or less, but intra-African relatedness is less than half that (more than twice as far away genetically).