The “words” of Jesus here are all of his sayings, commands, and promises. Collectively they are the “word” mentioned in v. 3 that makes us clean before God. This is all the words not only of Jesus in the Gospels but also of the prophets and the apostles since they spoke and write at the movement of Christ’s Spirit and spoke and wrote words about Christ.
These words of Christ “must so lodge in the disciple’s mind and heart that conformity to Christ, obedience to Christ, is the most natural (supernatural?) thing in the world.” (D. A. Carson, John, 517). Now we see why this is much more than mere obedience, mere outward conformity to a set of rules.
Jesus and the Word of God are so closely connected that they are interchangeable. The Bible is the written Word of God which testifies to Christ who is the Living Word of God (John 5:39). That’s why the Bible itself is said to be “alive and powerful (Heb. 4:12).
So this verse is telling us that one vital and important way that Christ abides in us is through His Word. And as we grow in the knowledge of His Word it begins to permeate our thoughts, our feelings, our choices, our very souls, just like nourishment from the vine flows through to each branch. The vine gives life to the branch this way and this is how our vine, Christ, gives life and spiritual vitality to each of us, too.
When we live in that power of that spiritual vitality brought about by the Word and communicated to us through the Spirit, we begin to bear fruit, not the kind of fruit that we think or that the world thinks is necessary, but the kind of fruit that such a divine branch would produce, fruit that is pleasing to God.
This is the context in which we must understand the promise about prayer at the end of v. 7. How does this happen? Obviously you have to be coming into regular contact with the Word through reading it and hearing it preached but the way the Spirit of Christ connects that Word to us is through prayer.
When we have such a vital, life-giving connection to the Word through prayer, we develop new hearts that desire only what God desires so that we pray and God truly gives us the “desires of our hearts” (Psalm 37:4) but this only comes after we have learned to so delight ourselves in the Lord that we desire only what He desires.
It is only as we are drawing life and spiritual vitality from the vine that we become not merely outwardly obedient, but develop such an inward connection to the life-giving force of the vine that it is like we aren’t even having to think in terms of obedience because we are made to desire the very things that Christ desires.
When I do what I want to do I don’t think of it in terms of obeying myself, I just do what I want. When the life of Christ flows through you as a believer, you will live in obedience to His commands, but you don’t even think of it as being obedience so much as you simply have learned to desire what Christ desires for you.