why do people argue?

Kitty: "What is the use of arguing? No one ever convinces another."
Levin: "Yes, you are quite right, for the most part, people argue so warmly only because they cannot make out what it is that their opponent wants to prove."

Levin had often noticed in arguments among the most intelligent people that after expending enormous efforts and an immense number of logical subtleties and words, the disputants at last became conscious of the fact that the thing they had been at such pains to prove to one another had long ago, from the very beginning of the controversy, been known to them, but that they liked different things and were disinclined to mention what they liked lest it should be attacked. He had experienced the fact that sometimes in the middle of a discussion one understands what it is that one's opponent likes, and suddenly likes it oneself, and immediately agrees with him, when all proofs become superfluous and unnecessary. Sometimes the reverse happens; one at last mentions the thing one likes, for the sake of which one has been devising arguments, and if this is said well and sincerely, one's devising arguments, and if this said well and sincerely, one's opponent suddenly agrees and ceases to dispute. This was what he wanted to express.

She wrinkled her forehead, trying to understand. But as soon as he began to explain she understood.

Kitty: "I see: one must find out what one's opponent is contending for, what he likes, and then one can..."

She had completely grasped and found the right expression for his badly-expressed thought. Levin smiled joyfully: he was so struck by the change from the confused wordy dispute with his brother and Pestsov to this laconic, clear, and almost wordless communication of a very complex idea.

-From Anna Karenina

Tolstoy's discourse on arguments is in the context of Levin and Kitty's courtship - how they started to communicate with very few words. Don’t let that undermine the message he has conveyed. Although most of us understand the 'anatomy' of an argument subconsciously, it is very useful (and important) to bring this up to the conscious mind. Arguments will always be part of our life, be it professional or personal. Understanding what the argument is all about, rising a bit, just an inch, above the situation, diffuses it much more effectively than breaking heads arguing just for the sake of it, just to satisfy our egos. I’m sure courting couples and the married crowd will be able to identify with this more than anybody else.

Now for some fun, go watch this:
More Cowbell, Skit on Saturday Night Live (SNL), will turn you into Will Ferrell and Christopher Walken's fan, if you aren't already.