In my opinion, this society became so violent because it became dehumanized. People stopped caring about each other -- they stopped seeing one another as human and so people became "fair game" for violence.
When you look at this society, you see that people do not care about each other. For example, Montag and his wife are really not at all in love. Millie's friends don't seem to really care about their families either. Families like the McClellans are completely rare.
If you stop caring about other people, what is to stop you from being violent towards them? I think that you will look at people and see them as objects -- just things for you to treat however you like. Once you see them in that way, violence towards them is no big deal -- it's just another form of fun.