Disinterestedness is fairly straightforward. One is a disinterested party in a matter when one's self-interest is not at stake. Moral jugments are disinterested judgments of approval and disapproval. They are made without reference to one's individual self-interest. Part of the answer is that people are capable of sympathy for others.
Human beings are capable of experiencing others' pains and pleasures vicariously.
This moves them to sympathize and hence to care. Sympathy thus operates to extend our approval and disapproval to cases in which we or our interests are not directly affected and about which we would otherwise not be concerned. In Hume's view sympathy is always operative, even when our own interests are directly affected. But at the same time or later we will also be motivated by sympathy to abstain from such action. Sympathy does not always win out, but we can at least see the matter from the sympathetic point of view too.