I should know this, but, what's the truth about how blacks in America were given the right to vote?

Question

And why was slavery abolished finally?

I mean what led to the changes? Why did they change?

Answer 1

The Fourteenth Amendment made African-Americans citizens and protected citizens from discriminatory state laws. Southern states were required to ratify the Fourteenth Amendment before being readmitted to the union. The Fifteenth Amendment guaranteed African American men the right to vote. Meanwhile, the Thirteenth Amendment abolished slavery.

http://www.thomaslegion.net/reconstruction.html
http://thomaslegion.net/aftermath.html
http://thomaslegion.net/causes.html

Answer 2

In addition to what the other responder wrote, the Voting Rights Act of 1965 removed the last barrier to black voting, which was a state-imposed "literacy" test which wasn't given to white voters as well.