This saying, that was found in a fortune cookie, seems to holds truth. Much like many of the sayings found in fortune cookies, if you meditate on it long enough, you will find that it holds truth.
When you are teaching another person, you have to focus on what you know. Focusing on what you know sharpens your understanding, and your mastery of that topic, therefore refreshing what you know, and teaching yourself more about it.
This came true to me when I was faced with teaching other people small amounts of ASL. I have taken five semesters of it, and that point I was the only student who wasn't on their way to becoming an interpreter. Three more classes and I would have probably been ready to take the test. But anyway, when you learn a lot about something, you find that you forget the basic things. When teaching the basics to some friends of mine, I had to stop and actually think about what the sign was, what handshape, and what direction it was going in. It surprised me that I forgot. But, when teaching others you learn what you forgot.
I think this saying also means that when you are listening to your student, and they are listening to you, there you learn how other people think. In that setting knowledge is flowing from one person to another, and you are surrounded by wisdom. Also, listening to others can be a great way to expound what you know.
But, above all else, silence can teach the most.