All Tybalt sees is that he is a Montague and he has been taught to hate the entire family since birth. This hatred is not productive and by hating Romeo and the rest of the Montagues, it does not help Tybalt in any way. If anything, it has a negative impact by being distraught and completely delirious. Shakespeare conveys here that nothing good comes out of blind hatred, there is no point of putting time and energy into blindly hating a person (or people) because it does not help you or anyone else.
Shakespeare continues to create even more tension between the two renowned families. Even after so many years, neither family has yet realized the devastating effects of such strong hatred. Once Mercutio and Tybalt are killed, both families get a sense of how much of a negative impact it has on each family. In one particular scene, Benvolio, Mercutio and Romeo are all together, and Tybalt arrives. Tybalt proposes a fight, and challenges Romeo, who calmly rejects. Mercutio instead decides to fight Tybalt. The both of them are dueling for fun, not meaning anything but just a friendly fight. Tybalt accidentally stabs Mercutio and kills him, which angers Romeo to kill Tybalt. As Mercutio is dying, he cries "Ask me for tomorrow, and you shall find me a grave man. I am peppered, I warrant, for this world. A plague o' both your houses!" (Shakespeare 3.1.103-105). Mercutio realizes how horrible this rivalry really is in his last breaths. He finally sees the impact it has on both families and how aimless and trivial it truly is. He says "both houses" in reference to the Capulet and Montague House, and how their feud is purposeless. Shakespeare depicts here that it took Mercutio losing his life to realize how futile blind hatred was. He shows here that blind hatred comes at a high cost and can be fatal. .
The reason why Mercutio and Tybalt even began to fight was because they were from either side.