Just as it is difficult to walk and chew gum at the same time, so it is confusing to visualize two views of the sphere on the same plane. The main reason for doing so is that it is a lot less work to draw. The problem is that clearly this got in the way of understanding.
At the end of the lab, we tried out an explanation with side-by-side views.
First, here is a crude 3D view that is a parallel projection from an odd angle. Notice the coordinate axes and a horizontal circle and a vertical circle, both of which are part of the sphere through I, J, K and with center O.
Now here is the side view that tells you for a point P on the sphere and in the (x,o,z) plane, what is the stereographic projection P' of P. This is a point on the X-axis.
Next we see a view from above of the same figure. The distance OP' is the distance from O for any point P at that latitude, so the circle of latitude through P on the sphere is projected to the red circle in the (x, y, 0) plane.
There is an interactive StereoDuo Java Sketchpad Page that shows both views simultaneously (plus a crude version of the 3D view). Since you can drag P in that version, it should give you a better sense of how this works for all P.