"You are already clean because of the word which I have spoken to you. Abide in Me, and I in you. As the branch cannot bear fruit of itself, unless it abides in the vine, neither can you unless you abide in Me. I am the vine, you are the branches. He who abides in Me, and I in him, bears much fruit; for without Me you can do nothing. If anyone does not abide in Me, he is cast out as a branch and is withered; and they gather them and throw them into the fire, and they are burned. If you abide in Me, and My words abide in you, you will ask what you desire, and it shall be done for you. By this My Father is glorified, that you bear much fruit; so you will be My disciples."
Jesus' apostles were already spiritually "clean" (i.e., cleansed from their past sins). This was made possible through their hearing, believing, and obeying the words of Christ (cf. Eph. 5:26). To remain in a "clean" state one must bear fruit by continuing to hear, believe, and obey the Lord. This is how one abides in Christ.
"I am the vine, you are the branches. He who abides in Me, and I in him bears much fruit" - Here Jesus plainly declares to His disciples that they, as individuals, are branches attached to Him (and therefore dependent upon Him). Those who abide in His word in order to obey Him are His disciples (cf. John 8:31)--such will bear much fruit.
"For without Me you can do nothing" - Outside the vine, no branch can bear fruit or even live; outside of Christ, no disciple can be productive or even survive. Let it be understood that all metaphors, if their details are pressed to an extreme degree, will break down. For instance, physically it is absolutely impossible for a branch to produce any fruit if it is not attached to a vine (or tree). Spiritually, however, it is possible, to a certain degree, for some spiritual fruit to be produced by non-Christians (e.g., Cornelius, Acts 10:1,2). In fact, some spiritual traits (or fruit) must be present in order for one to even become a Christian (e.g., love, faith, self-control, etc.)! One must understand that Jesus' fundamental point here is that God wants good fruit, made possible by obedience to the Scriptures, produced in the lives of His disciples. His point here is not that only Christians can produce such fruit, but that we must remain faithful to Christ if we intend to please God and be saved in the end.
The consequences of a branch (i.e., disciple) being cut off from the true vine are twofold: (1) spiritual withering and (2) the fires of hell (cf. Matt. 25:41). Before physical death (and Jesus' return), one may repent of his sins and be forgiven. In so doing he will be reattached to the vine and will receive spiritual nourishment.
To "abide" in Christ includes more than just being in His spiritual body (i.e., the church); it requires living in close communion and fellowship with him. One who does not fully submit to the divine will does not abide in Jesus. Those who truly "abide" in Christ and allow His words to "abide" in them have the promise that their desires will be granted. This is the case because those who have given themselves to the will of God will not request that which is forbidden or selfish; they will petition the Father for those things that will lead to more fruitfulness, and the Father will grant such.
"By this My Father is glorified, that you bear much fruit; so you will be My disciples" - Fruit-bearing is proof of faithful discipleship (cf. Matt. 7:20). When disciples of the Lord bear fruit, they reflect honor not only on their Savior but upon the One who sent Him into the world (cf. Matt. 5:16). When one does good works and is conformed into the likeness of Jesus, he will produce blossoms of "love, joy, peace, longsuffering, kindness, goodness, faithfulness, gentleness, and self-control" (Gal. 5:22,23).