Answer: It reinforces the idea that without equality in the US, the idea of a free society is a joke.
In this excerpt, Douglass wants to highlight the hypocrisy of arguing that the United States is a country that enjoys equal rights for all. He thinks it is foolish to claim this when Black Americans do not have the right to vote. Douglass repeats the phrase "without this..." in order to reinforce this idea, and highlight all the limitations that Black people have in society.