In some ways, the feud that exists between the Montagues and the Capulets can be compared to a strong rivalry between two schools. The rivalry could be academic or athletic. It really doesn't matter, because the end result of the rivalry is the same. You and your school are given someone to hate. Rivalry games are often the biggest game of the year, even if the game isn't likely to be a good game. It's simply fun to hate the other team so much. We love our rivalries because they instill a sense of passion, pride, and family within us. It's us versus them.
The Capulet and Montague feud is a rivalry between families. Readers aren't given a long, drawn-out explanation as to why the rivalry exists. We just know that both families are wealthy and influential families in Verona. I personally suspect that the rivalry stems from some kind of economic or political power play that went badly between the families. What is clear is that the families hate each other, and a character like Tybalt gives us the impression that he likes the feud. He is strong-willed, passionate, and argumentative. He's also extremely loyal. He loves being a Capulet. He loves to protect the family. It brings him honor. Fighting against the "evil" Montagues is how he shows love for his family.
The feud gives Tybalt and other characters a target to hate as well as a target to love. The emotions come hand in hand, and they can't really be separated from each other. Consider happiness for a moment. Is it really possible to know that you are happy if you have never experienced sadness? Love and hate are two sides of a different spectrum. The characters know that they love their own family because they hate the other family with equal fervor.