Slavery in any part of Africa didn't just begin. Slavery, sadly enough, has been a part of many cultures of the world for thousands of years. Not only has it been a part of it, but it was more or less socially accepted as well. What I mean by this is that when one tribe went to war with another, then the victor would conquer the enemy and make their defeated foes a part of their spoils.
With that part said, let's go back to early exploration by the Spanish and Portuguese. Hundreds of years before Columbus set sail to the Americas, Europeans had tried to find a shorter water route to Asia. The task was difficult simply because the technology was poor so developing structurally sound ships on a consistant basis was hit or miss. Navigational tools by the Europeans were less than adequate to handle long distances as well. As mentioned before, Spanish and Portuguese were constantly pushing the limits by sailing further and further south along the coast of Africa. With schools being open by people such as Prince Henry the Navigator of Portugal, more and more sailors learned and dared to find a way around the southern tip of Africa.
As these sailors sailed along the coast, coastal docking stations were made in various points on the western coast of Africa to fix ships, get supplies. Inevitably these docking areas were marked and claimed by certain countries, particularly the Spanish and Portuguese. By the mid 1400's better maps were created to pass along to others for reference and use as to where to stop for safety, supplies, help, etc... These areas were developed and contact was made with the local tribes of the areas such as those of the Kingdom of Songhai. Goods were exchanged as was typical when cultures collide.
It just so happen to be at this same time the Spanish and Portuguese were exploring the west coast of Africa and laying claim to certain coastal ports, some were toying with the idea of exploring across the Atlantic to the west for a shorter route to Asia. The idea was nothing new matter of fact, it was just now there was more interest. We all are pretty much familiar with Columbus' discovery in the Caribbean by the late 1400's and how he claimed there was more gold in the area. He did this in part so he could get more funding from the King of Spain to keep exploring.
The natives he came in contact with would suffer from the claim of gold in the new world. Many new conquistadors would come in search of this gold and the natives were pressed into slavery. So much so they would literally be worked to death to the point many tribes would be wiped out to extinction in the gold mines. Who now could they use for the mining and menial work? Remember what was mentioned about the west coast of Africa?
Explorers who were already on the west coast of Africa were now being propositioned for accessing manpower. Tribes who were already in contact with European explorers in west Africa were fully aware of how this could play into their favor. As the saying goes, "The enemy of my enemy is my friend." The slave trade along the coast was an unfortunate lucrative business for all involved. Consequences be damned, but profits speak louder.
In closing, the slave trade on the west coast of Africa was a culmination of events that had horrific and sad results. Profit was the ultimate catalyst as to why it continued.