Not a historian but as an enthusiast I believe I can answer your question.

Over much of Europe abolition became popular, as they themselves did not depend on slaves like the Sugar colonies or the Southern States of the US they had no qualms with ending it, besides the fact that many of them sold slaves it wasn't a large enough factor to break the movement.

The British Empire, the most powerful force at the time, especially in the Atlantic used its naval dominance to stop all trading of slaves. Britain banned the slave trade in 1807. It did not BAN slavery but banned the enslavement of new slaves and the trading of slaves.

So tl;dr: Humanitatian reasons, a lot of intellectuals in Europe and parts of America were against it and managed to convince their legislatures to ban its sale.

An important reason must have been the Haitian Revolution, in which Haiti became a self-governing state of former slaves.

The Atlantic slave trade of the eighteenth and nineteenth century went mainly to the Caribbean, where the conditions were much harder than in the mainland of the Americas, and thus more slaves died without giving birth to or raising a next generation. But surely the Haitian Revolution made Caribbean slave owners think twice about the number of slaves they held.