I got a little confused by this slide in lecture; how exactly does the computed surface intersection end up under the surface itself? Is it because of how the light source and camera rays line up? And how does this exactly result in the shadow ray interacting incorrectly?

it is due to numerical precision issues - when the intersection point is super far away from the origin of your world's coordinate system, then the "resolution" decreases since the step between adjacent floating point numbers increases. therefore, the closest one to the real intersection point might be underneath the surface. this is a problem because we calculate shadow rays from the point of intersection, and if we self-intersect then that point will be treated as being completely in shadow even when it is not, leading to the black specks