In the proof above, why do the boundary curves of $ S_1$. and $ S_2$ need to be oriented in opposite directions? Why are the boundary curves of each respective hemisphere oriented in opposite directions? How do you prove the flux of the curl or the two hemispheres are equal?
On
The boundary of any manifold is always orientable. We may choose a normal vector pointing into the interior of the manifold (or alternatively one pointing out). For any point on the equator, the normal vector pointing into the interior of the upper hemisphere is opposite the one pointing into the lower hemisphere.
You don't have to prove that the integral of the curls on the two hemispheres is equal. That is an outcome of this problem, not an intermediate step needed to prove the result.
The problem proceeds by first using Stokes' theorem to show that the the two surface integrals are in fact, equal to the two line integrals on the curve $C$ of F. Then the solution is pointing out that regardless of what value $F$ takes on that curve, one of the terms is the negative of that other because they are the same function integrated over the same curve, just in an opposite direction. Hence they cancel.
To answer your first question about the boundary curves are oriented in opposite directions, use the right hand thumb rule for the orientation (look at the figure in the section "a rotating body" to see how to hold your hand - if thumb is along the normal, the direction of your fingers gives the direction of the boundary). The outward normal in the upper hemisphere points up, but in the lower it points down, so the boundaries are oriented in the opposite directions. Also are this Khan Academy video explaining this.