When Act III Scene I opens Benvolio and Mercutio argue as Benvolio suggests they should leave because if they encounter the Capulets they will not "'scape a brawl," and Mercutio accuses Benvolio of being "as hot a jack in thy mood as any in Italy." This small argument foreshadows his heated conflicts to come later. But, surprisingly, it is Mercutio, who has cautioned Benvolio, that immediately flares up when Tybalt accosts him. Further, Mercutio's anger then ignites Tybalt's; so, when Romeo enters, and surprisingly, declares that he now loves Tybalt, Tybalt is insulted and becomes incensed. Romeo's declaration of love is also perceived as "calm, dishonorable, vile submission" according to Mercutio, who then draws his sword. Most surprising of all, though, it the fact that Romeo's peaceful intentions backfire and end up causing Mercutio his life. Then, Romeo suddenly chastises himself for his "womanly" peaceful attempts and draws his own sword, killing Tybalt.
One point of surprise is Mercutio's whirlwind of action addressed to both families as he curses them both. Mercutio is a friend of Romeo's but a relative of Tybalt's and of the Prince. He curses both families for their vain glorious feud and, indirectly, the Prince for failing to control his kinsmen and subjects.
Mercutio's death came as a surprise to me when I first read Romeo and Juliet, because he was such a compelling dramatic character in the play. I really thought that Tybalt was going to back down, and I especially did not expect Romeo to snap the way he did and kill Tybalt after Mercutio's death. Like post #6 pointed out, Romeo acts very rashly in this scene, completely giving himself over to his emotions, which we all know proves to be a very fatal character flaw in the end.
Given that when we first meet Romeo he is pining over Rosaline and then goes off into the woods to be alone, it seems rather unusual for Romeo to be in the middle of a fight which ends in death. Romeo goes from moping around to seeking revenge and killing someone. This is yet another example where characters in this play behave rashly and do not think about the consequences of their actions. Interestingly, although the Montagues and Capulets are adversaries, we have not seen any other place where Romeo gets involved in a fight. Here, he ends up killing someone--a member of Juliet's family. This scene changes the entire direction of the play. Had Romeo not been banished to Mantua we can only imagine what would have happened between Romeo and Juliet.
The previous posts are good answers. I would add that this scene always surprises me a little bit because of Romeo's boldness. I always think of him as a lover, not a fighter, but he certainly becomes a fighter, and a good one. It shows another side of Romeo.
By definition, the climax of the play is the point at which a decision is made or an action is taken from which the main character cannot go back to the way things were before. In this play, Romeo's decision to kill Tybalt is the climax of the play because it is what drives the rest of the conflict and action of the play to its ultimate conclusion. If Romeo had listened to his thought that through marriage he is technically now related to Tybalt, he wouldn't have killed him, wouldn't have been banished, and he wouldn't have come up with the crazy plan to be with Juliet. The play would have been entirely different. It is a suspenseful scene because it leaves the audience wondering what will happen now that Romeo has made such a fateful decision.
I think it's because of two things. For one thing, it is the turning point as mentioned above. For the second thing, this is where there is the most action (as opposed to talking) in the play. This is where the dramatic swordfight happens. Because of this, it is the most action-packed scene in the play.
The scene is the turning point of the play. It is when Mercutio is killed by Tybalt, and Romeo kills Tybalt. As a result, Romeo is banished, and the audience can already sense that things probably aren't going to end well from there. Mercutio's death is probably the most surprising moment in the play, as he really has little to gain by fighting Tybalt, who insists that he has no quarrel with him.