Not only don't I know how the field functions, but I don't know how the machine was able to view the target and how it knew which target should be shown.
What I really don't understand is why the machine is even necessary. Kirk is the Captain. He can order any crew member anywhere. All he has to do is order an enemy to the transporter room where his henchmen club the guy on the head, and then he can beam the remains into space using a wide dispersal pattern. Why all this magical technology just to kill men, which men have been doing since the dawn of time?
Well, as said, Kirk commands resources. One of those would be CCTV feed of his own starship; the monitoring aspect is solved there and then. Surely that's more elegant and effective than the "all he has to do", needlessly complicated and labor-intensive alternative of ordering one of his underlings to spy on the victim?
Another resource he commands is the transporter. Again, surely more elegant than going through the complications of first killing, then transporting, or relying on henchmen?
Mirror Kirk's power lies in his ability to do things his competitors cannot. Those who employ henchmen and phasers in dark corridors are the hopeless underdogs, as shown when one of them makes a move against the Tantalus-protected Kirk. Push-button murders give Kirk an edge, both as sheer tactical expediency and as the ultimate in plausible deniability ("I was in my cabin, there's no blood in my hands or lingering charge on my phaser, and you can't even be sure he's dead since there's no body").
The only mystery bit here is why it takes "an unknown alien scientist and a plundered lab" for Kirk to get to possess this ultimate assassination tool. Wouldn't the integration of ordinary starship assets into this death machine be a thing the
least likely to be achieved by an outsider?
But if we disregard the obvious line of speculation based on what we see and instead go by what is stated, then we accept that an alien machine fell on the lap of Kirk and Kirk only. Perhaps it indeed combines perfectly ordinary transporters and surveillance techniques, but does so without tapping into the corresponding starship resources, where Kirk's competitors could notice the tapping, interfere with it, or do their own? An alien machine could also work outside a starship, even if what we see never requires us to think the Tantalus Device actually can. Or perhaps its alien superpower is its ability to defeat all those security measures everybody in the Mirror Universe has in place already against surveillance and transporter assassination? Even then, it might merely be a better and more powerful type of transporter/camera combination, triumphing by taking a leap in the rat race.
As for improvements, the obvious one would be making the device portable and easily concealed on Kirk's person, giving Kirk the ultimate smiting power. As matters currently stand, he's reliant on at least one henchman most of the time - and the more trustworthy that henchman, the more likely he or she is to ultimately backstab him, by the social rules of the Mirror Universe!
Timo Saloniemi