What’s the key thing that interviewers are looking for?

Interviewers fundamentally want to know whether you’ll be interesting to teach. “Supervisions”, or “tutorials”, are what makes the teaching at Oxbridge unique. A focus on regular, small group teaching is the hallmark of an Oxbridge education. Your interviewers therefore want to know how well you engage with discussion in this kind of setting.
Act interested: they want to see that you’re excited by your subject. Ask questions: demonstrate that you’re open to developing your own understandings. Read broadly around the subject but stake out a niche: if you can evidence in depth knowledge about a specialist area sometimes interviewers will let you lead the conversation. If you don’t always have an answer, don’t worry: humility can be a positive, and stating that you will go away and explore the idea further shows an eagerness that your interviewers will like.