How did slaves mobilize public opinion to achieve goals? Why was it important to gain that general public opinion?
I would argue that this question is based on a flawed assumption. It assumes that the slaves in the American South were active in mobilizing public opinion. This is simply not true. The slaves did not have the means to effectively mobilize public opinion in the way that, for example, the Civil Rights Movement did in the 1950s and 1960s.
In order to mobilize public opinion, you have to have a way of getting your message to the public. For the slaves in the US, this would have been the public in the North since the Southern public was pretty solidly in favor of slavery. Slaves in the South would have had very little way of getting their message to the North. They would not have been able to write letters, pamphlets, or anything else. They would not have been able to get interviews with media outlets. They simply had no way of getting their message out.
Instead of talking about how the slaves mobilized public opinion, I would say that the abolitionists mobilized public opinion (or tried to). The abolitionists lived in the North and had a variety of ways to get their message out. Abolitionists like William Lloyd Garrison and Frederick Douglass gave speeches. Abolitionists like Douglass and Harriet Beecher Stowe wrote books. Garrison published his newspaper, The Liberator. By doing these things, they were able to get their message out to the public in the North.
It was important for the abolitionists to get public opinion on their side so that they could pressure the US government to do something about slavery. The only way to get rid of slavery was through government action. The government was not going to do anything about slavery unless it was pushed to do so by public opinion. This is because the government would have to have the political will to override the strong opposition from the South to abolish slavery. If the abolitionists wanted slavery abolished, they would have to mobilize public opinion so that there would be enough pressure on the government to force it to act.
As it turned out, the abolitionists never really got public opinion to be solidly on their side. Instead, it took the Civil War and the anti-Southern feeling that it created to convince the US government to abolish slavery.