If there's a TL;DR for this one, it's location, location, location.

The African continent, like most regions of the world, had systems of unfree servitude or outright bondage in the 15th century; part of this is because (per John Thornton's work, but also that of others like John Iliffe in general terms) power over people was the crucial thing. Land was plentiful; knowledgeable human beings to work it were less so, and thus captives taken in war, people sentenced for crimes, or otherwise forced to labor in recompense (and depending on the society, sometimes people born into bondage) were widespread across a variety of African societies. A trade existed by simple dint of extension, especially for male war captives, and when a seaborne outlet became possible then the removal of problematic people could be attractive. This trade, however, was really quite small until almost the 17th century, certainly compared to what it would be later and even what it was internally to the continent. So why the change?

In the beginnings of the Atlantic world, a variety of unfree people saw transportation and use as laborers in various places, for fixed terms or indefinitely depending--some Europeans, lots of Native Americans, some Africans, and perhaps a few others. Who was used where often depended on logistics and needs. In the case of West and Central Africa, the issue was one of sugar islands and plantation farming.

See, when the Portuguese began traveling to West Africa in the 1430 and 1440s (and yes, that was the initial destination; India overtook it only after the first 60 years), they quickly found that human beings were among the cargo available. Usually, the best prices in early years remained along the coast; merchants would trade fine goods to the Portuguese (and other Europeans who rapidly started to horn in on trade despite Papal sanction) for such people, in what was known as the coasting trade. By this coasting trade, shipowners and investors could make significant wealth, especially in places like the Gold Coast (today southern Ghana) where societies demanded a lot of labor to mine gold and clear forest. The other potential use was on the islands off the coast of Africa. Any island significantly away from the African coast, with few exceptions, was uninhabited; most of these were volcanic, fertile lands. This was a perfect place to extend sugar growing that Europeans had begun to adapt from Islamic systems in the Mediterranean, and the island of Sao Tome (sorry, can't do the special characters on this keyboard) quickly became the center for sugar growing for much of Europe. Thornton and Curtin (Rise and Fall of the Plantation Complex) note that Sao Tome was responsible for the plurality, if not majority, of Europe's sugar for much of the 16th century. The backbreaking toil of sugar plantations--where growing, processing, refinement, and shipment all happened--initially fell to people who were a mix of Europeans, Africans, some Asians (though poorly documented), and pretty much anyone the Tomistas (upper classes of Sao Tome) could get. The problem was that, with the African coast so close, goods and replacement laborers increasingly came from there. African endemic diseases established on the island, which further diminished the viability of other laborers. The Tomistas--along with other Portuguese--also fomented civil war and raiding around Kongo (the western DRC, Angola, and Gabon) in order to generate captives, and supported various rulers who promised not to cut off the flow. This established a connection between oceanic slave transportation and plantations that the Dutch West India Co. (the 1621 iteration) would regularize and extend in a systematic way to Brazil, and when eventually booted there, its managers filtered out into the Caribbean and spread that system. The same problems in Sao Tome established in other tropical climes--especially yellow fever, malaria, and other tropical illnesses to which African populations had some resistance--which further encouraged that system.

In effect, the Atlantic trade was the extension of an existing traffic, which itself was an extension of a smaller traffic, each expansion of supply and demand (and change in the meaning of the people so enslaved) building on the last. In the case of the Americas, too, the idea that enslaved people couldn't escape because of their skin color and lack of relationship to Native Americans or Europeans was common, but in reality escape and refuge were quite possible. What made it harder was the increased brutality of the system itself, especially as plantation labor (for sugar especially) depended on expending the enslaved person as a unit of labor instead of keeping that person healthy and well. By that time, in the mid 1600s, it was far cheaper simply to get another captive person, unless the person in question had special skills (such as blacksmiths). Other plantation crops, and systems of management, were not nearly so murderous in environment or demands--but they still depended on the same equation of people with units of labor, like beasts of burden or machines.

So although European ships took some enslaved Africans to the Americas in the 16th century, where they toiled in mining or other capacities, it was not common to the point of institutionalization. African plantation slavery only became prevalent, and then institutionalized, across the tropical Atlantic in the 17th century, as the whole system spread west. By that time, shifting racist ideas about Africans and the inertia of practice grew together with ongoing supply from intermediaries on the continent to cement the equation of slave labor with people of African descent.

Thornton's (now aged) Africa and Africans in the Making of the Atlantic World 2nd ed (Cambridge: CUP 1998) and A Cultural History of the Atlantic World (Cambridge: CUP, 2012) are good places to start, which will provide more grain than I give in this very oversimplified description. Phil Curtin's Rise and Fall of the Plantation Complex (1996?) is another, and there are a few more but I'm not with my library right now. I'll see if Paul Lovejoy's Transformations in Slavery 2nd ed (2009) talks about the extension; I believe it does. This was a complicated mercantile system with very complicated origins, but it really did have a lot to do with the particular proximate relationship between enslaved African labor and sugar planting. Of course, as if to stick another finger in the collective African eye, the Indian Ocean planting system that came of age after the end of the 18th century would also employ this proximity and many of these ideas to pick up the global slack as the Atlantic export slave trade became less viable in the decades after British abolition. But that's another story.