What happens
Two Capulet servants — Sampson and Gregory — pick a fight in the streets of Verona. Two Montague servants take the bait. Benvolio, a Montague, draws his sword to break the brawl up. Tybalt, a Capulet, draws his to keep it going. Both households' patriarchs arrive and try to wade in; the Prince stops the fight and warns that the next person to disturb the peace will pay with their life. The square clears. Romeo enters last, miserable about a girl named Rosaline who has sworn off love.
Why it matters
The play's whole world is set up in twelve minutes. We learn that the feud is so old it's fought by servants who don't know its origin, that the state has lost patience, that Tybalt loves a fight, that Benvolio tries to stop them. Shakespeare gives you the entire political situation in one street scene — and the cost of getting it wrong is announced ahead of time by the Prince. When the fight in Act 3 Scene 1 comes, every sword-stroke is the consequence of the speech we just heard.
Romeo's first appearance is a small masterclass in misdirection. The audience has just watched a city-wide brawl. Romeo arrives alone, in love-poet mode, complaining about a girl who's sworn off men. Shakespeare opens the romantic plot in the wrong key on purpose. Romeo's sighing about Rosaline is so studied — the oxymorons, the Petrarchan paradoxes — that we register him as a young man performing love rather than feeling it. The play will spend Act 1 teaching him the difference.
Dramatically, Shakespeare also plants the play's metaphors here. Brawling love. Loving hate. Smoke and fire. Cold fire and sick health. The vocabulary of paradox that Romeo uses for Rosaline becomes, by Act 5, the literal vocabulary the play lives in: love that brings death, peace that arrives too late. Act 1 Scene 1 is the play's whole future, said in advance, by people who don't know it.