The Atlantic slave trade was one of the largest economic ventures and humanitarian tragedies of its time, if not in all of modern history. Its rise was the result of several factors mostly occurring in the Americas during their colonization by European powers.
The Spanish and the Portuguese had been bringing slaves to the New World since the 16th Century. However, in the 18th Century, the British ramped up the kidnapping of Africans and enslavement of them in the Americas on a scale previously unknown.
As Europeans settled in the Americas, they established large agricultural plantations. These enterprises, particularly sugar plantations, made European businessmen very rich and drove up demand for their products in Europe. They also required a large workforce to maintain its operations.
European colonists had at first planned on using the indigenous populations as a source of forced labor. In many places, particularly the Caribbean, North America, and Brazil this proved ineffective as the native populations quickly succumbed to European diseases. A new source of cheap labor needed to be found.
The British in particular turned towards indentured servants as a cheap source of labor. However, by the early 1700s, the supply of indentured servants could not keep up with demand. Not enough Europeans were signing up for this arrangement. That's when the British looked in earnest at Africa as a source of forced laborers. Pretty soon, a thriving and barbaric trade in Africans began that made even more profits for the merchants involved than before.
Another factor that should be considered is the racial attitudes of many Europeans at the time. Europeans were able to justify their treatment of Africans with their belief that white people were superior to other races. Views that black people were naturally suited to physical labor, unintelligent, and in a perverse way benefited from their enslavement allowed countless slave traders and plantation owners to allow slavery to flourish in the New World.