In order to understand what hegemonic discourse is, you first have to understand the term "hegemony." This is a term that is typically used by people who are influenced by Marxist ideas.
Marx (and others like Gramsci) believed that the dominant classes in a society have many ways to keep the other classes and groups down. The best of these is hegemony. Hegemony is a means of maintaining social order without using force. If hegemony is achieved, the oppressed classes obey the dominant classes not because they are forced to, but because they believe that it is right to do so. In this way, the oppressed classes "participate in their own oppression" because they believe that they deserve to have less power than the dominant classes.
Hegemonic discourse, then, is a way of talking about things that supports hegemony and makes it seem as if the current social order is inevitable and natural. In the context of anthropology of development, a hegemonic discourse would be one that makes it sound as if the developed nations inevitably and naturally should be on top and that any development in other parts of the world must proceed along lines that are acceptable to the developed world.