West Africa (where most of the slaves in the Americas came from) already had long standing traditions of 'slavery', although they were nothing like those systems that the Europeans and Americans would create. Europeans could easily purchase African slaves from other Africans along the coast. Africans were eager to sell their neighbors because they didn't really think of themselves as some kind of monolithic 'culture' in the way that Africans are usually grouped by people today. And again, Africans had been making servants out of one another for generations, there just were no moral obstacles to selling each other off to Europeans. The Europeans of course offered them valuable trade goods; such as firearms. Firearms would give a group an enormous advantage over their peers, so you can imagine they were highly sought after. It was a very favorable arrangement as far as most Africans were concerned, although in the long term it definitely depleted manpower from the region and caused a very unstable political organization, which made it very easy for European colonialists to dominate the region years later.

As for WHY African slaves were used... The Europeans needed labor for their endeavors in the Americas, whether it be mining or doing work on plantations. They used the natives at first, but their weakness to European diseases and the harshness of their treatment decimated their populations and thus the labor pool the Europeans could draw from. There was some advocacy at the time to stop the atrocities against the natives, such as from Bartolomé de las Casas, but in the end they still needed labor.

Europeans were both unwilling and unsuited to the work in the Americas, as most of the labor requirements were in tropical climates. Europeans used to temperate environments usually did not fare well when forced to work in the tropics, and they were susceptible to disease there as well. Africans were both well adapted to life in a tropical, or at least hot, environment AND had immunity to the Old World diseases that the Europeans carried which had so decimated the natives of the Americas. The relative ease with which African slaves could be acquired, Europeans unwilling to satisfy the labor needs themselves, and Europeans capable of ignoring the obvious moral wrongs of slavery for economic gain, and it made perfect sense to use African slaves enmasse.

I think there are a few misconceptions about the slave trade in Africa that need to be addressed here. I'll be using examples mostly from central Africa. Slaves from central Africa were headed mainly to Portuguese territories (Brazil), most American slaves were from further north. That said the processes in place have some similarities.

The need for slaves in the economies of the Americas comes from the sizable amount of labor needed in a plantation. European citizens were unwilling to leave Europe to work in the plantations of the Americas. In a lot of European states domestic slavery was frowned upon. Slavery was Illegal in England, I'm afraid I cant talk for other European states in that regard. However, the profits to be gained from American plantations were too great to be missed. So slaves exported from Africa simply filled an Economic niche for the Europeans.

Its implied in the question (and a couple of answers) that Europeans had a sizable military presence that they used to control African populations and enslave them. For the most part this is untrue. For Europeans on the continent the slave trade was a economic affair, European traders attempted to source and negotiate for slaves rather than enslave them themselves. Initial contact with Africans would mean an integration into a much wider economic sphere. Before European (and Arab in some cases) contact African economic systems were limited to trading nearby. Archaeological evidence seems to suggest the main traded items were, Gold and Copper luxury goods and Iron tools. Slaves too were probably traded moderately. Post European contact the goods expanded to the exportation of goods to the coast in return for European luxury goods (cloths spices etc) and guns that couldn't be gained by Africans.

Each individual African polity is slightly different in how trading with Europeans operated but the most successful (during the slave trading period) operated through a system of royal monopolies. The King or Chief of an area would own a monopoly on all trade then distribute the goods of the trade vertically down to his subordinates. This meant his subordinates had a reason to be loyal to him and to get more of whatever good they were trading. This in turn meant that the King was reliant upon the long distance slave to ensure the loyalty of his subjects and the stability of his state. So what we see is in the 15th to 19th centuries is some African polities developing in a way as to export as many goods to the coast. The majority of these goods would eventually become mostly slaves.

An examples of this sort of structure is the West Lunda Mwata Yamvo state. The West Lunda were deep in central Africa in the modern day Katanga region. They conquered nearby states integrating them into the state and taking slaves. The Lunda then traded these slaves with intermediatarys such as the Ovimbundu. Intermediary peoples such as the Ovimbundu specialized in the transportation of goods (including slaves) to the coast. Unlike the Lunda these states were not expansionist states but developed a merchant society as a result of the slave trade. The slaves and other goods would then be transported to either intermediary states (such as Kasanje who forced all traders to go through them) or directly to the coast where they would be exported to the Americas.

Initially in the 16th century this was small scale, however as demand for slaves increased, supply increased to meet it. African expansionist states enslaved more people and raided for slaves specifically in weaker nearby states. The caravaning states increased the size of their caravans to compensate. Each of the African actors had its own reason to export slaves.

However, this is not a case of, Africa was technologically inferior and therefore could not defend themselves from being enslaved. Right up until the end of the 18th and start of the 19th century the balance power lay with African rulers. Europeans were often denied entrance to the interior (to protect royal monopolies from being circumvented) and could be executed if they insulted leaders. So what we see is a two factor system that enabled the slave trade to become so huge: The demand of Europeans and the unique political systems of African polities to fulfill the demand.