What are you expecting as an answer to your question? "Because" is the usual answer to a "why" question. This is similar to other languages. For example, in Spanish (as far as I know), "por qué" starts a "why" question, and "porque" starts a "because" answer.
Now, maybe you know that "because" answers "why". However, you also noticed that "Because we want you" is not a complete sentence, and that is the real intent of your question. Or maybe it isn't. I'll answer that question too, anyway. The complete sentence that means the same thing as "Because we want you" is "I didn't go see him because we want you." The because is still there, it is just a complete sentence now. In informal (usually spoken) English, people don't try to follow rules as much as they try to make sense, and that makes complete sense: "Why x?" -> "Because y."