To me that means that if the source of a magnetic field is on one side of a super conductor, the other side should have no effect from it.
The physical mechanism for the expulsion of magnetic fields from the interior of a superconductor is the generation of surface currents which produce a secondary magnetic field, cancelling the field in the interior. There's no particular reason to think that they will cancel any part of the field in the exterior region as well, and in fact they don't.
You may be making the mistake of assigning more physicality to the field lines than they deserve. Remember that the magnetic field is a vector field - an arrow attached to every point in space. The magnetic field lines are a convenient way of visualizing the field, but you must be careful not to assume that they behave like physical objects do. In particular, your illustration indicates that you feel that they should behave like the trajectories of massive objects, which they do not.