Jesus started another analogy by comparing Himself to the vine and the Father to the vinedresser. Jesus also compared the disciples to the branches. The branches produce fruit with the vine but can't bear fruit by itself.
In fact, Jesus talked about how if a branch does not "abide" by the vine, it is essentially casted out and gets withered. Such branches eventually get burned (15:6). We know that the disciples must have accepted Jesus (as evident by Jesus's comments in 15:3) and that the disciples must be eternally secure (as noted by Jesus in John 10), so what does this mean? The short answer is that the "burning" references the discipline and accountability that rogue believers would have to face. After all, fire is a common image used to refer to the accountability people will have to God, and even believers can face such fire (as noted by Paul in 1 Corinthians 3). Grace in Focus has a more detailed article in this page, and Gary Derickson talked about the analogy as a whole in this article.
But what's more important is learning how to have a "fruitful" life, and the key is abiding in Jesus. Jesus says that when the disciples abide in Jesus, they can ask their desires for such desires to be done -- and it will be done (15:7). Note that when one abides in Jesus, their motives and focus line up with Jesus's motives and focus. In fact, as we abide in Jesus, we become more compelled to do or ask what Jesus wants. As a result, the Father is glorified, and the disciples can bear lots of fruit (15:8).