Christ was the "true light" in John 1:9 and the "true bread" in John 6:32. Here He is the "true vine." He Himself produced fruit for God when Israel, as a nation, had not (e.g., Isaiah 5).
Believers are called the branches, so this chapter emphasizes how we also can bear fruit for God. Vine wood is not really good for anything but producing fruit---you can't build anything with it, for example. As branches, we are really only designed for pleasing God through fruit-bearing; and that comes through our connection with the vine.
The Father as well as the Son are involved in this fruit-production process. If the branch is not producing as it should, there is activity to help it bear fruit. And if it is already producing, there is further activity to increase the yield.
There is a one-time aspect of the cleansing (verse 3), and there is also an ongoing pruning or cleansing (verse 2). [Note that the verbs in these two verses are really the same Greek word in the original New Testament writing.] What security there is in knowing that God is carefully and tenderly looking over the believer to help each one bear fruit!
Verse 6 describes what might be called a "false branch." The Lord is not speaking about His disciples there; He does not say "you" as in the previous verses, but talks about someone else in general terms. This verse declares judgment on one who only appeared to be a branch ("cast out as a branch," or like a branch), and there is burning reserved for this false disciple. Others are involved in carrying out this judgment (perhaps like the angels in Matthew 13:41-42), but not the Father or the Son in their caring character of this chapter.
For the real branch, the key to fruit-bearing is given in the words of Christ Himself: "Without Me you can do nothing." Branches must abide in the vine if fruit is to develop. This is an aspect of our Christian lives which we have some responsibility for, although it does not mean our salvation depends on us.
"Abiding" is explained in the following verses. His words, His love, and His joy all become a genuine part of our Christian lives.
The branch that "abides in Me" is showing dependence. Branches that are independent from the vine will produce no fruit. "And I in Him"---our fruit increases when we carry Christ with us, so to speak, in our daily lives in this world. We are still here, but we have Him in us. We show His character. He can restfully "dwell in your hearts by faith" (Ephesians 3:17).
Even further, when His words abide in us and we enjoy the communion that comes from asking and receiving from Him, there is much fruit (15:7-8). The word of God and prayer will cause us to abide in Him.
Fruit is for God to enjoy. The Father is glorified when we bear much fruit.
Fruit in the Christian reproduces the life of Christ. Fruit is not a set of good works on their own merits, but in fact every word and every action of ours will be just like His. It is the natural experience of the believer; we are branches in the true vine, so we should bear fruit.
More of that is produced by less of self.