Answer by Christopher Lamke:
This is an interesting question. Others have answered it well, but I feel like weighing in to see what I think about this.
I'm going to use this definition of:
"an insulting way of thinking or behaving that comes from believing that you are better, smarter, or more important than other people"
I think the question stems from a misunderstanding of atheism. As it says in my tagline, "Atheism is a lack of belief in god(s), nothing more." That may be hard to understand or accept, but it's true. A lack of belief in god(s) is no different in principle than a lack of belief in unicorns or werewolves. There's zero objective (observable, physical) evidence for the existence of god(s) or other supernatural beings, and that includes the gods of Christianity, the Greeks, or any of the many other well known pantheons.
As a believer, you should be able to understand that your belief in only one god or set of gods, while rejecting the gods of all the other religions, is based on something other than objective evidence. The rejection by countless other people of your god(s) and all other gods beside their chosen one(s) indicates that the belief in god(s) is based not on objective factors but on factors like culture. Around the world, people are very likely to follow the religion of their family and local culture. To assume that all these other people, from widely varying cultures, are all wrong and you are right, is problematic. In fact, it generally fits the definition of arrogance above.
There are atheists who are arrogant, who don't believe in gods but also believe that it's foolish or stupid to believe in god(s). This is a subset of atheists and places them in almost exactly the same position as theists who accept their god(s) and reject all others as foolish or stupid.
As far as not needing a god to exist, it seems inappropriate to call this stance arrogant because it's trivially true and whatever your religion, the majority of the world lives well without believing as you do. For many thousands of years, people have lived praiseworthy, full and productive lives without believing in god(s). Atheists are only slightly different than followers of traditional religions, in that they lack a belief in one more religion than each follower.