What is an Oxbridge interviewer looking for?

2 main things:

Intellect – your intelligence: This includes showing that you are able to think on your feet, able to logically and clearly articulate a view point and defend it and able to adapt your point of your view to different situations. For instance, the interviewer's objective is to push your viewpoint to the limit and see how you cope with this. 

A passion for the subject - The people interviewing you are experts in their field and have very often written the books that students study from. So they want to teach someone who is just as passionate and enthused about their subject as they are. So you need to convince them of that (without referring to the term "passion" of course!) For example, even if you were contemplating between law and history: as far as they are concerned, you have always wanted to be a lawyer. In particular, make sure that you can distinguish between the practice of the subject and the academic study of the subject. Academics don’t want to hear that you want to go on to practice your subject e.g. to become a lawyer or teacher because they would like nothing more than for you to complete your Oxbridge degree and become an academic. So for example, make sure if you’re answering a question like, ‘Why law?’ you don’t say because you want to be a lawyer after your degree, keep it very focused on the academic study.