In John's Gospel, Jesus tells his friends that he is a vine and they are the branches. Some Old Testament prophets, like Ezekiel, had used the image of a vine to talk about Israel; they had been put in their land by God in the same way that a gardener would plant a vine, choosing the right place for it and taking care of it. But Jesus says that he is a vine - or rather, he says that he is the true vine. God the Father is the gardener, but it's Jesus, rather than the nation of Israel, that is the vine.
Anyone who chooses to follow Jesus, he says, is a branch of the vine. God, the gardener, will prune the branches to make them more fruitful, or will cut off the branches that don't bear any fruit. This seems rather worrying - I often don't feel like my life is very fruitful: does this mean God will cut me off and throw me away? - but Jesus goes on to say that anyone who remains in him will bear much fruit. I might not feel fruitful, but I'm reassured that remaining as a branch of the vine is all that's required for fruitfulness.
In fact, Jesus doesn't only say in verse 5 that the fruitful person remains in him; he says that he, Jesus, also remains in that person. So being fruitful (achieving your potential, or being the kind of person you were meant to be) doesn't only involve you staying in Jesus, but also him staying in you.
