A species is considered invasive in an ecosystem if it has migrated there from another ecosystem, has no natural enemies or effective compeititors, and is therefore able to have disproportionate success in its new ecosystem—to the detriment of biodiversity in its new environment. Successful invasive species cause chains of extinctions in their ecosystems, sometimes completely changing the environment.
The Argentine ant (Iridomyrmex humilis) in North America is a canonical example. Armies of Argentine ants have successfully wiped out both competing ant species and ant-eating predators in many of the areas they've colonized. Notably, North American Phrynosoma lizards can't eat them, and subsequently starve to death when their usual food sources have been wiped out by the invaders. In fact, Argentine ants are a globally invasive species; they have established a global megacolony that covers every continent except Antarctica. An example of an introduced and harmful species that does not meet the usual criteria for invasiveness is the small population of Central American freshwater turtles in some Danish lakes and ponds: They can cause a lot of damage to the local ﬁsh and amphibian populations for a while, but since they don't breed in the cold climate, they're a temporary nuisance, not a serious ecological threat.
Consider now the waves of large animal extinctions in the last few tens of thousands of years: Almost the entire megafauna of Eurasia, Australia and the Americas died off with spectacular rapidity, following patterns of human migration. The few species that survived into historical times were reported to either have no fear of humans or to have completely ineffective tactics for dealing with humans—because there was nothing even remotely like humans in their native ecosystems. How would they have any chance to evolve effective defenses against highly-intelligent, tool-using, cooperating primate long-distance runners? Some Eurasian and Oceanian megafaunal extinction events actually occurred during Homo erectus migrations!
In contrast, much of the African megafauna has survived to the present day. Most of it is either instinctively afraid of, or aggressive towards, humans. Furthermore, tropical Africa is host to a very large number of horrible diseases (and disease-carrying insects) that are so effective at killing large populations of humans that pre-colonial tropical African societies deliberately avoided having large concentrations of people in areas close to water, to make outbreaks controllable. Humans evolved in Africa, and the African environment evolved to cope with humans.
In other words: Outside Africa, humanity is a textbook example of an invasive species.
Much of our global success as a species (and why human populations outside Africa got an early leg up in the civilizational race) is a direct consequence of this.