The very short answer is obvious: A lot.
As I am writing this paper on CAS and queer theory, I keep thinking about this. My paper is very much about how certain aspects of queer theory help us think through certain issues in CAS. Queer theory does more than simply make visible an abject subject (though it certainly would do enough if it was all that it did), it also provides insights, tools, and methodologies for other fields of research and inquiry. So, I am trying to think through some of the things that CAS can offer to other lines of inquiry, besides simply bringing forth and giving weight to a repressed subject (that of course never being a 'simple task'). For example, I think CAS has allowed me to think through the questions of the biopolitical. The many problems I have with Agamben aside, I think he gets right that the biopolitical is rooted in an anthropological machine. Critical animal studies gives us tools for understanding that anthropological machine, and hopefully tools for dismantling it, as well. Moreover, CAS has given me the ability to understand the central importance of the issue of opacity. I think I have often talked about how wonderful I think Glissant's book, The Poetics of Relation, is. One of the major themes in the book is the political affirmation of opacity. CAS gives us ways to begin to think of ethics and politics with beings who obviously maintain their opacity.
There are many other areas (I am pretty tired right now), but I think we will shortly be seeing more and more research projects that uses CAS to think through and extend projects in other fields, at the same time those fields will continue to influence critical animal studies.