I occasionally have meetings with my supervisor. When the meeting finishes, my supervisor says: "cheers". I don't know how to answer that. What is a natural response to that? What do native speakers say in this situation?
The best answer is "Cheers", followed immediately by whatever you'd like to say, such as "Thanks" or "See you on Tuesday" or "That was really useful".
You then need to go through the goodbye ritual, which you can instigate by saying "Bye!" or something similar. They are trying to signal the end of the meeting!
Don't ask them another question—unless it's important!
When we have been conducting a long conversation or social interaction, it is difficult to decide when to finish the conversation. How do you know, for example, when the other person is ready to finish. You certainly don't want to extend the conversation much past this point because it is difficult and annoying for the other conversant and can cause frustration.
One way we have of solving this problem is to have codified pre-closing signals. These are not the same thing as saying goodbye, they are a signal that the utterer is ready to to finish the conversation. These signals require a response from the listener, which informs the leave-taker that they have recognised the conversation is finishing. This closing signal is normally followed by a valediction, in other words by saying goobye. Native speakers of a language subconsciously understand and use closing signals. They are therefore difficult to teach, unless you have actually studied them.
In phone calls, by far and away the most frequent closing signal is ok. There are many that we use in face to face situations, for example, alright, right, ok and so forth. Cheers is used as a friendly and informal initiation of leave-taking. The use of cheers as a leave-taking device has caused problems for people who try to study valedictions in corpora, because it can be used in several ways, as a toast, a pre-closing signal, a valediction and an expression of gratitude.It can be hard to distinguish the expression of gratitude, for instance from the valediction (consider, for example, a speaker who says cheers before leaving shop).
Leave taking rituals can cause problems for language learners, because they may be slightly different in different languages [see bottom of p. 7 here]. In particular, there may be no straightforward translation from the user's L1. To see how complex this can be from the other perspective here is a page on the pragmatics of Spanish leave-taking strategies
You can read up a bit on some language-specific and universal features of leave-taking rituals here. [It's free but you need to set up a J-stor account - takes a minute or so]