Matthew's Gospel records a single scene: Jesus tells Simon Peter he is the rock the church will be built on, then hands him the keys of the kingdom of heaven, with the power to bind and loose. Centuries of Christian tradition built an entire doctrine of papal authority on those few verses, reading Peter's commission as passing, office to office, down to every pope after him.
Perugino's fresco makes that argument visible rather than merely quoting the text behind it. Christ does not simply speak to Peter; he physically extends two keys into Peter's open hands, one gold and one silver, a detail read since the Middle Ages as spiritual authority paired with earthly authority, the power to govern souls alongside the power to govern the visible church. Peter kneels rather than stands, receiving rather than seizing, which is itself part of the argument: authority conferred, not authority claimed.
The Vatican Museums' own account of the fresco reads the scene as exactly this, the conferment of powers onto the first vicar of Christ on earth. Commissioned for the pope's own private chapel, painted at the center of its north wall where every cardinal gathered for a conclave would face it, the fresco was never simply decoration. It was an argument in paint for the authority of the man who commissioned it, and of everyone who would sit in his chair after him.