How does Romeo insult Tybalt?

To answer this question, take a look at act 1, scene 5. In this scene, the Capulet party has just started and Tybalt notices that Romeo, a Montague, is present. For Tybalt, Romeo's attendance is an insult because he comes from the family of their enemies. More importantly, Tybalt believes that Romeo has come to mock them, as he tells Capulet:

A villain that is hither come in spiteTo scorn at our solemnity this night.

Tybalt is so angry about Romeo's "intrusion" that he wants to fight and kill Romeo. Capulet, however, does not want any fighting at his party. This prompts Tybalt to leave, but he will not forget Romeo's presence at the party. In fact, he fully intends to make Romeo pay later on:

I will withdraw, but this intrusion shall
Now seeming sweet, convert to bitterest gall.

This quote foreshadows Tybalt and Romeo's next meeting, in which Romeo does not want to fight but ends up killing Tybalt in revenge for Mercutio's murder.