As Paul says, even though we as human beings know God, we refuse to acknowledge him. That's what Peter did. He refused even to "know" Jesus! Peter's failure reflects all our failure. It forces us to face the reality about ourselves. But the point of the story is that Jesus foretold this - he knew it was coming. And Jesus forgave Peter, when Peter confessed his love for Jesus. So the story illustrates both the horrible nature of sin, and the amazing reality of grace. That's essential to the whole meaning of the gospel.
Consider the generosity of our Savior: what He acquired by dying becomes ours by eating. As often as we receive this Sacrament with proper dispositions, we make our own the fruits of all the labors, injuries and sufferings of His life, especially those borne at the time of His passion and death. Just as the power and the sensations of the head reach all the members of the body, in the same way, because Christ is "the head of the Church which is His Body" (Eph. 1:23), the treasures of His grace are made abundantly available to all who through charity are one with Him as living members.