Why is it required for Test Administrators and Proctors to constantly walk around the room during testing?
It is not globally required: e.g. I do not move around the room unprompted when proctoring an exam.  So it is probably a matter of local academic culture and/or preference of the (head) course instructor.

In these kinds of fine points, academic culture differs so strongly from place to place as to make it clear that there are not always deep, well-thought out reasons for doing things one way or another: many institutions do things in a certain way because they "always" have done so.  The academic culture shock that I experienced in moving from the US to Montreal was considerable.  

For instance, I believe that I first learned the word "invigilator" when someone introduced himself to me as an invigilator for my course.  If you had told me before I arrived in Montreal that final exams there have professional invigilators, I might not have believed you.  In the US, if you are teaching a course without a "common final" (I imagine that common finals occur only in a small minority of all courses taught, but I don't know for sure) then you, the instructor, are very likely the sole inviligator/proctor/administrator/TA in the room when an exam is taking place.  (Not that having professional invigilators is a bad idea: if you have the infrastructure in place, why not?  But I assume that they get paid at least a little bit, and this would never fly in the state university at which I currently reside.)

So understand that when I say that I do not move around the room when giving a final exam, I mean no one does.  It is not really clear to me what cheating I might be missing out on by only looking at the students from the front of the room.  I should say that I am used to a classroom environment with stadium seating and small flip-up desks so I can see a lot from the front of the room.  I also teach a subject -- math -- such that coming in with a cheat-sheet or something like that would not make things that much easier.  Even consulting the internet on your cell phone while in the bathroom would not help that much -- one does need to show one's work, after all -- unless the student was unusually insightful and well-prepared about how to cheat in this way.  In my local academic culture, cheaters are not insightful or well-prepared...