It worked through the action of literate people that supported the cause, although they were not a big group until the second half of the 19th century. If I'm not mistaken what really changed and helped the abolitionist movement in Brasil was the Paraguay war fought between the allied powers (Brasil, Uruguay and Argentina) against Paraguay. Instead of fighting in the war themselves, several slave owner families sent their slaves to fight in their places, which made them gain some relevance in the society.

One element that contributed heavily to that was the the soldier slaves were considered as "brothers" by other soldiers (whose origins were from the poor white tiers of Brazilian society in that period, in opposition to the rich slave/land owners from the south-southeast regions of the country), fact that made the republican movement stronger in the second half of the century. It was not a strong movement whatsoever, but it was functional and there were se relevant members of society that were engaged in the movement.

Pressure from England also contributed to that, enacting several laws regarding commerce and trafficking of slaves in the period. This came as a result of a necessity of having people to buy industrial goods from the empire (remembering that england was going through its second industrial revolution). Of course it wasn't a direct transition, because from one day to another former slaves wouldn't be able to have enough resources to become important buyers in the modern world-economy. There is a collection called "História do Brasil Nação" (I don't know if it's available in English, I'm sorry) that explains really well the century's history and all of it's internal and external actors.