What is the context?
The vine-and-branches picture makes the relationship clear. Jesus is the source of life, and His disciples depend on Him. The promise of much fruit is joined to remaining in Him and His remaining in them.
The surrounding passage explains what this dependence involves: His words remain in them, they keep His commandments and they remain in His love. The image describes a continuing relationship that shapes their life and work.
The final statement about doing nothing apart from Him concerns the fruit of that life. It is not a denial that people can perform everyday tasks; it presses the disciples to recognize the source of the life He is calling them to live.
How is this passage commonly understood?
Christians commonly read this as a call to rely on Christ for spiritual life and fruitfulness. Calvin on verse 5 emphasizes that the power for such fruit comes from Him. The Catholic Church’s Catechism, paragraph 2011, also teaches that Christ’s love and grace are the source of good actions. The shared dependence should not be obscured by treating Christians as though they generally believe they can produce this life alone.
Where does this fit in God's plan?
Galatians 5:22–25 describes love, joy, patience and the other fruit of the Spirit, and calls those who live by the Spirit to walk in step with Him. This gives practical shape to the fruitful life Jesus describes: God forms His people’s character as they live in dependence on Him.
What is our position?
Our study Prayer and Faith describes remaining in Christ as living in keeping with His teaching and example. Dependence on Him shapes how we live and what we desire; it is more than adding religious words to plans we have already decided to pursue.