What a wonderfully existentialist question. I have to say however that I think the question is moot. Without teachig society would crumble in a generation. The objective of teaching is to teach, to transmit knowledge and skills from one person to the other. You might as well ask "why do we raise children."
Anyone who has taught also knows that unequaled feeling of intense caring and satisfaction of knowing we have helped someone and prepared them better for the world. True teaching is a priceless gift that can be given again and again into eternity.
Of course in the age of compulsory education, and in those instances of it before this age, we do see education foisted on some of the unwilling, but I don't think they are unwilling to learn other things. I believe curiosity and the desire to know (unless it is rooted out of us) exists at the core of every person.
Some people exploit teaching to try to indoctrinate, or force belief or belief systems on others, but I think that's a totally different question altogether.