President Johnson was speaking to how black citizens are equal to everyone else and should have the same rights as everyone else. He wanted them to have the right to vote and to have a voice. They were not the ones causing trouble. They just wanted to be treated fairly. It was the people and the old outdated laws they followed that caused the inequality issues.
Some slaves bought their own freedom from their owners, but this process became more and more rare as the 1800s progressed. Many slaves became free through manumission, the voluntary emancipation of a slave by a slave owner.