"Remain." Remain means "continue - in this verse it indicates faithfulness. There are some translations that use the word "abide" instead. Abide - is an interesting choice - because it includes notions of obeying and dwelling.
Basically - if you remain with Jesus in the context of this verse - you obediently continue with Him - not for a weekend retreat - but you unpack your bags so to speak. You're with Him for the duration.
The beginning of this chapter Jesus says He's the true vine ... reminding us that there are some not so true vines out there too.
How can we ever know if we're abiding in the true vine?
When we abide in the true vine, in Jesus, we bear fruit. What's fruit? It for sure is not popularity, or money, or power. When you're fruitful - there's more of God's Kingdom available for us all to live in. Then we help each other abide in such a way that grace, love and truth are easier to see - and just maybe - sometimes they're even a little easier to do. Fruitful sounds like humility and looks like caring for one another. Fruitful is what we're made for - and frankly - it's so awesome that Jesus is dying for us to know what that's like. Ahem.
Terrible puns aside - imagine someone mentoring you - and being so committed to that role that they give you the key, fling open the fridge, and let you have your own room. You'd know that mentor was committed to you ... and the only thing that would be left would be for you to ... well ... accept it.