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What did Jesus entrust to Peter?
Start a few lines before the keys appear. Jesus asks his disciples who they say he is. Simon Peter expresses his faith in Jesus as God's Son and Messiah. Only then does Jesus speak to Peter about the rock, the Church and the keys. Reading the exchange in that order changes how we see a painting of Peter. Before he is the man who holds the keys, he is the disciple who recognizes Christ. Matthew 16:13–19, NABRE
An earlier biblical scene helps explain why a key can stand for authority. In Isaiah, Eliakim is entrusted with the key of the house of David. Opening and shutting are part of the responsibility he receives in another's household. The USCCB note on that passage points readers directly to Matthew 16:19. It is a good image to keep in mind if the keys make you picture Peter stationed outside heaven deciding who gets in. The Gospel scene concerns a charge Jesus gives him among the disciples. Isaiah 22:20–22 and the note on verse 22, NABRE
Catholics also read the scene in light of the other apostles. The Second Vatican Council describes Peter as the bearer of the keys and explains that the authority to bind and loose is also given to the apostles together with him. That helps explain why a Catholic image can single out Peter even when the other disciples are standing beside him. Lumen Gentium, section 22
A portrait and a handover tell different stories
Look at the painting at the top of this page. Cover the lower half for a moment. There is an older man looking upward, his face caught in the light. Uncover his hand, and two long keys give you a reason to call him Peter. Art historians call an identifying object like this an attribute.
The National Gallery of Art traces this figure's pose to a painting of a different apostle, Philip. The artist could borrow a pose and still change whom we see by changing what the man holds. The National Gallery's account of Saint Peter
Now look at the foreground of Perugino's fresco in the Sistine Chapel. Peter kneels before Christ. Follow the keys from one figure's hand toward the other's and the emphasis shifts. The portrait asks who this man is. The fresco asks who gave him this responsibility. Perugino makes the promise visible as a handover. Matthew records what Jesus says without describing a pair of metal keys passing between them. See Perugino's Handing over of the keys at the Vatican Museums
Why are the keys crossed on the Vatican flag?
On the white panel of the yellow-and-white Vatican flag, two keys cross beneath the papal tiara. No one is holding them. Here the keys serve as an emblem of the papacy, recalling Peter because the pope is his successor. Matthew speaks of keys in the plural but gives neither their colors nor their arrangement. You can recognize the biblical reference even when a painter places both keys upright in one hand. Vatican City State's guide to the flag
The next time a child points to keys in a church window, try asking, “Whose hands can you see?” One man holding them gives you a clue to his name. A second figure offering them invites you to tell the Gospel story. Keys displayed on their own invite a closer look at the flag, seal or shield around them. If the glass is damaged or you can only see part of the scene, look for an inscription before settling on an answer.
If that conversation leads to why Catholics put pictures of saints in churches at all, our guide to statues and sacred images takes up that question.