This is a fascinating piece of art that makes me wonder how the cut out was created.
Can anyone explain to me, in layman mathematical terms, how the position and angle of the black stick, when rotated, relates to the intersection of the plane (cut out)?
Mike Miller's comment is exactly to the point, so I am repeating it.
The slot cut in the card is in the shape of a hyperbola. If you imagine the card rotating around the axis instead of the disc, the hyperbolic slot would trace out a surface called an elliptic hyperboloid. This surface is ruled, which means that it can be decomposed into a union of straight lines. The straight bar passing through the slot indicates a different member of this family of lines at each instant.
This animation from Wikipedia makes the whole thing clearer: https://upload.wikimedia.org/wikipedia/commons/thumb/5/59/Cylinder_-hyperboloid-cone.gif/220px-Cylinder-hyperboloid-_cone.gif