Saint John's gospel uses the agricultural image of vine and a vine dresser to express a relationship that is unique. Quite singular when you think that neither the Jews nor the Muslims would admit in terms of intimacy between the Creator and creature, Father and Son, God and me. So, why is Christ called the 'true vine' and why are we his 'branches'? The short answer is because it is our Christian belief, our Christology, that God is waiting for humanity to bear fruit, sin notwithstanding. The Incarnation, and the proclamation of the Good News tells us of the wine of love, obedience and prayer with the goal of uniting God and humanity in a truer way.
That we are expected to "bear much fruit" and to rely on the Lord for all things there is a hope that we remain in Him and that His "words remain in you". There is a dependence on God in a radical manner that is unheard of in most of relationships. To remain, to abide, to stay close to Jesus is the key of the spiritual life. Not to remain in Christ is reject the offer of Grace. The question of what it means to remain in Christ is given by the second reading: keep the commandments, of both Testaments of sacred Scripture and the teaching of the Church. Concretely, we are nourished by Christ Himself in the sacraments of the Church, notably in the Holy Eucharist.