The abolitionist movement did not directly accomplish very much. It did not, of course, lead to the peaceful abolition of slavery. The slaves were freed, but it is hard to argue that the abolitionists were the direct cause of their freedom.
We can say, however, that the abolitionists had an indirect effect. That is, they helped to bring about the Civil War and the Civil War was what freed the slaves. Ironically, though, the abolitionists brought the Civil War about more by their impact on the South than by their impact on the North.
The North did not go to war in order to abolish slavery. In that sense, the abolitionists did not cause the North to go to war. However, the South did go to war largely to preserve their “freedom” to have slavery. They felt, in part because of the presence of the abolitionists, that the North wanted to end slavery. Their fear and suspicion of the North was a major cause of the Civil War.
Thus, the abolitionists accomplished something, though probably not what they wanted to. They helped make the South fear the North so much that it opted for secession and war. This war brought about the end of slavery, which was what the abolitionists wanted.