A symmetrical airfoil in level flight has a positive (greater than zero) angle of attack. is: The airfoil may be symmetrical, but it coes not encounter the airflow symmetrically. The airflow patterns around a symmetrical airfoil with a positive angle of attack are very similar to a cambered airfoil, with a greater amount of air flowing over the "top" of the air foil, at a higher velocity than the air flowing underneath.
A symmetrical airfoil at a zero angle of attack will produce no lift, but that ain't the same as level flight.
Could never get a good answer for this one out of the books!
Haven't been reading the right books then. Off the top of my head, I think that "Aerodynamics for Naval Aviators" gives a clear explanation of this, although I don't have a copy in front of me at the moment. Many other books on aerodynamics also explain this, including diagrams and photos from wind tunnels using smoke streamers to show airflow.
There's nothing mysterious about this at all.