Thinking through someone else's mind is something we understand. For example, you decide not to make a particular dish for your dinner party because one of your guests wouldn't like it. You're thinking about the dish through someone else's perception of it.

Cothinking is like making this effect bidirectional. The other person is induced to think things through the frames and constructs you are giving them, yet they feel they are doing their own thinking. At the same time, in order to employ this effectively, you must have a deep understanding of how they think.