What happens
Paris comes to the tomb to scatter flowers on Juliet's grave. Romeo arrives with Balthasar and a crowbar. Paris recognises Romeo as Tybalt's killer and challenges him. Romeo tries to send him away, then, when Paris insists, kills him. He grants Paris's last wish and lays him beside Juliet. Romeo speaks his final words to Juliet's body, drinks the poison, and dies. Friar Lawrence arrives too late. Juliet wakes. The Friar pleads with her to leave; she refuses; he runs. Juliet kisses Romeo, finds his dagger, and stabs herself. The Watch arrive. The Prince, Capulet, and Montague gather over the bodies. The Friar confesses everything. The two families shake hands.
Why it matters
Shakespeare structures the scene around timing. Romeo arrives minutes too early — the Friar's potion was timed for him to be the first to wake her. Friar Lawrence arrives minutes too late — long enough for Romeo to have drunk poison, not long enough to keep Juliet alive. Friar John, with the letter explaining the plan, never arrives at all. The play has been counting half-hours since Act 1, and Act 5 Scene 3 is what it adds up to.
Juliet's death is one of the cleanest decisions in literature. She wakes, sees Romeo, hears the Friar tell her to come away, and refuses. The Friar runs. She tries the poison on his lips — none left for her. She kisses him. She finds the dagger. She uses it. The girl who said "I'll look to like" in Act 1 is the woman who decides this in Act 5, and the play is careful not to soften the speed of her choice.
The reconciliation that follows is, deliberately, not a happy ending. Capulet and Montague shake hands over their dead children. The Prince's last lines name the cost: "All are punish'd." The grudge is over because the people who carried it have nothing left. Shakespeare wrote a love story that ends in a peace agreement made on top of a tomb, and didn't pretend either thing made the other one all right.