What obligation to "the opinions of mankind" does jefferson percieve in the Declaration of Independence?
From the way that you have phrased this, it sounds to me as if you are simply asking for the specific words that Jefferson writes, not for the general meaning of what he says. I will, however, answer in both ways.
If you are just looking for the exact words of the Declaration, the part that you are looking for comes when Jefferson says
…a decent respect to the opinions of mankind requires that they should declare the causes which impel them to the separation.
The obligation that Jefferson perceives, then, is the obligation to “declare the causes which impel them to the separation.”
Let us now look at what this means. At the beginning of the Declaration, just before the part quoted above, Jefferson says that he is talking about a situation in which it has come time for people to cut their ties with a country to which they have allegiance. He is talking about situations like that of the colonies, which wanted to break away from the country that they were a part of.
So, what Jefferson is saying here is that people who want to break away from their country must respect the “opinions of mankind” and let mankind know why they feel that they have to break away.