The Atlantic slave trade was not something that arose overnight in the form that became famous. Instead, it developed gradually over a number of years.
The trade can be said to have started with the Portuguese. Their explorers made their way down the coast of Africa in the 1400s. They made contact with Arab merchants who procured slaves for them from the sub-Saharan kingdoms. These slaves were taken in relatively small numbers and were used for domestic service in Europe and for some agriculture on islands off Africa and Europe. Eventually, the Europeans started to get their slaves directly from West Africa as their trading routes expanded and as they came to need more slaves to work in the Americas. This slave trading was done mainly by Spaniards at first as the pope had given them the exclusive right to trade with the New World (this was before the Reformation so everyone essentially obeyed).
At this point, the Europeans started systematically trading for slaves. They built trading posts where they kept slaves that they had bought until ships could take them away. The Europeans traded with coastal kingdoms for slaves that those kingdoms’ people took from other states and kingdoms. This form of the slave trade was essentially the final stage. The only changes that occurred were in the identity of the traders. The English came to dominate the trade in the late 1600s and beyond.