We should remember the past because if we do not than we are in danger of repeating it. That is the main "why" I can give you. It is absolutely crucial that we see what mistakes we have made so that we cam learn from them and not do them again.
As to how, I don't think that we "owe" anyone anything at this point because none of us were even alive when the slave trade was happening. What we do need to do is make sure that we treat all people as equals with respect. We need to make sure, as a society, that all people have equal access to education and to job opportunities. We need to be ever vigilant to stop racism and to teach future generations that discrimination is not acceptable.
What we learn from the past, we also need to carry into the present, however, if we are truly going to say that we have learned anything. Today, discrimination seems to be more against Muslims (often viewed as terrorists because of a few extremists) and homosexuals (because they are viewed as living a different lifestyle from the "norm"). Discrimination against these two groups is not much different from the discrimination against African Americans in the past.
It has been said that those who do not learn from the past are doomed to repeat it. Why we must address the past is so that we don't repeat it, and how is with respect, understanding, and awareness of the potential for it to happen again.
I say we paid our debt to the slaves when we sacrificed our lives in the Civil War. They also had a chance to become free under Lord Dunmore's Proclamation of 1775. Over 5000 Slaves escaped that way.