We sang this often at BYU. So often that I had it memorized probably after one year at school. Overall it is a song about service. Yet I believe that was a huge way in which Christ taught--by serving them. Because it showed forth honest, sincere love for that person, and a sincere desire for that person's best interests (hence the inclination to teach).
This also made met think of the hymn just before that, #308.
The most important part of Jesus' role as a teacher was that He loved those He taught. (Which happened to be everyone.) Love is a vital element to have if one is to teach. Love for what one is teaching, and especially love for those whom one is teaching.