The Easter season is approaching its high point: not Easter day, but Pentecost! The coming of the Holy Spirit marked the beginning of something new in the world: people filled with God's living Spirit! And this people would be visibly united around a person who represented the very one who sent the Spirit; a Vicar of Christ.
There are a couple of references in the Gospel to a special appearance of the Risen Christ to Peter. In Luke (24), we see the disciples from Emmmaus (arriving with their story of the Risen Lord) greeted with "The Lord has been raised! He has appeared to Simon." St. Paul reminds the Corinthians (1 Cor 15): "Christ...was raised to life...and appeared to Cephas" ("Peter" in Aramaic). So John 21 could be an account of that special one-on-one of the Risen Lord with Peter.
What strikes me is that this passage, set on the lakeshore in Galilee, could have just as easily been describing that scene Matthew sets in Caesarea, in Matthew 16. These are the only two passages in the Gospel where Peter is addressed, solemnly, as "Simon, Son of John" (Matthew really says "of Jonah" but you get the idea).
In Matthew, Simon son of John is told, "Blessed are you!... I will give you the keys of the Kingdom." This is followed by an announcement of the coming sufferings of the Messiah and Peter's attempt to dismiss all that. To which Jesus responded forcefully, "Get behind me, Satan!" And his instructions: "If anyone wants to be a follower of mine, let him renounce himself, take up his cross and follow me..."
Doesn't that sound amazingly like today's story? "Do you love me? Feed by lambs... When you are older, someone else will bind you and lead you where you would not want to go. Follow me."
That "follow me"
In John, then, Jesus puts the keys of the kingdom in Peter's hands after a profession of his love. Whereas in Matthew, Peter "knows" Jesus, here it is Jesus who "knows all things," including Peter's love.