I think this answer has something to do with the thrust vector. I remeber this Q from a long time ago. Obviously written by someone with too much spare time on his hands.
Simply put, the thrust in level flight is horizontal. If not, in the climb for example, there is a vertical element. The vertical element generates a small upward reaction, therefore less lift is required in the climb and therefore less of an angle of attack is required.
If you have problems getting your head round this, consider an aircraft with more thrust than mass - i.e an F15. The pilot can maintain a vertical climb where all the weight is being supported by the engine thrust and none by the wings. Therefore the lift generated by the wings is zero and the anle of attack zero as well.