What does Mandela mean when he says for to be free is not merely to cast off ones chains but to live in a way that respects and enhances the freedom of others?

It means to be free and to let others also be free by respecting the differences in many aspects that we have.

Explanation:

What Mandela means, is that we can not be free if we do not let the other be free too, that is to say, that we have to respect each other above our differences and try to enhance our collective freedom to live a better life.

This is what Mandela did when he became president of South Africa by not punishing the ones that made him and his community suffer, instead he tried to free all the inhabitants of South Africa and taught them to respect each other and their differences to live a happier life without resentfulness.