In ancient Israel, vineyards ranked second in agricultural importance, just behind wheat. As Jesus often does he is using a common everyday image to springboard into his theology. There are seven I AM statements found in the gospel of John and in each case Jesus shows the disciples something which is right in front of them to see something hidden. This is one of the divine paradoxes, that the hidden nature of God is revealed in things already in front of us, therefore not really hidden at all.
There are two things that stand out to me in this section of John. First, Christ defines our purpose as Christians. Just like the vine, we are to bear fruit. In this context, the vine branches that do not bear fruit are useless and take valuable nutrients from the vine itself. They are therefore cut off and thrown away. So, we must stay true to our calling and work in all things to the glory of God.
Secondly, and perhaps obviously, a vine branch cannot exist alone. It must remain attached to the vine to live. There is great theological importance to this understanding because Christ did not leave the role of the vine ambiguous, but places himself as the vine. We, the branches, cannot live unless we remain connected to Christ, not to kingdoms, nations or institutions, but Christ. And, we find our connection to one another not through ourselves but through Christ. Jesus defines the type of interdependence that God instills in all of creation, that through God we live, and move and have our being.