This quote is from Act One, Scene Five of Romeo and Juliet and is spoken by Tybalt rather than Romeo. To understand why Tybalt is so angry here, we must acknowledge that there is a terrible conflict between the Montagues and Capulets that has gone on for longer than anyone can remember and which has unknown origins.
In this scene, Tybalt (a Capulet) spots Romeo (a Montague) from across the room and realizes that this enemy has snuck into the masquerade ball under the cover of a mask. Tybalt perceives his presence as an immediate threat to the celebration, and so when he says, "Now, by the stock and honor of my kin, / To strike him dead I hold it not a sin," he means that he wouldn't consider killing Romeo a crime since it would be an act performed to honor the Capulet family. Tybalt runs this plan by Lord Capulet, but is quickly rejected by the patriarch, who orders Tybalt to "Let him alone." This only inspires greater anger from Tybalt and a thirst for revenge that will later prove deadly....
When Tybalt says this (not Romeo), it is because Romeo has crashed the Capulet party and Tybalt recognizes him. Tybalt is saying that because the Montagues have insulted the Capulets' honor, it isn't a crime/sin to kill Romeo (also because Romeo is there uninvited, which is another insult, and perhaps because he's drooling over Juliet).
Tybalt says this not Romeo, but it means that by the honour of his family he would kill for his family. It also suggests that he shares a lot of emotion.