I generally resist analogies, but since you're a software-only guy, and since everyone else insists on talking to you about capacitance, I'm going to go for the story.
Imagine you are firing paint balls at a wall. Everywhere you hit the wall, there's a splat of paint. Except there's one place you fire at, and when the paint hits it, it makes a little mark but otherwise the paint just kind of disappears. When you investigate, you find someone has attached a suction device to the other side of the wall, and since the wallboard is paper, it's able to draw the paint right through it.
The mechanism in the phone is doing a similar thing. It drives a bunch of electrons onto the backside of the glass. Normally this can then be observed as a rise in voltage. But at the spot your finger is touching, the charge is absorbed by your finger, and the voltage at that location will be significantly reduced. How it works through the glass has to do with the fact that charges attract and repel each other, but you didn't come here to learn electronics, so we'll just be happy with this much explanation.
Perhaps you can see that a dime placed on the screen doesn't draw off enough charge to be registered by the touch circuits, but if that dime is connected to something larger, like yourself, now enough charge is drawn off to be detected. This is rather more like the behavior of static electricity, rather than the classic circuits involving batteries, wires, resistors, and light bulbs.
[Edit, to address, "What can I do..."]
I do not know if something like this can be made to work, but the goal is to be able to simulate a touch by coupling the wire over the screen back into the ground system of the phone. The grounding (done by laying the phone on a metal plate or sheet) is also capacitive and is the equivalent of holding the phone in your hand. The idea is that the transistor switch can disconnect the short lead over the screen, and that there wouldn't be enough capacitive coupling back to the controlling circuit (the computer you were going to try to generate touch events from) to trip it when it was supposed to be off. Switching the transistor on would connect the lead to the ground plate and should then generate a touch event in the phone.
[Edit: An associate says the stray capacitance of the FET would still be too much to let this work. Doesn't cost anything to try, though.]
[Edit: This part of the answer is in response to the comment found below]
It's not about the mass; it's about the surface area. A sheet of foil can hold more charge than a dime will, even though it weighs less. For a piece of wire, it's more about the length. Equally important is whether that wire is attached to anything. If you are holding it in your hand, then the wire plus your body is in the equation. Your body is as good as a piece of metal, in this experiment. And any metal, be it copper, nickel, aluminum, or steel will work. The only requirement is that it conducts electricity.
Incidentally, I just tried a few metal objects on my phone, and I'm surprised you are getting so much response from the wire. A dime does nothing unless I touch it with my finger. If I hold a quarter in my hand and touch the edge to the screen, I get nothing. Two doesn't do it either. Three, stacked sideways provides enough contact to affect the screen. Next, I tried a paperclip, held in my hand. It would not respond if the rounded end was touched to the screen. I had to hold it the long way and touch the side to get a reaction. A metal bar stood on end did not affect the screen, but touching the opposite end with my finger caused it to react.
[Edit: @toolbear asks about the "ground"]
Voltage is the difference between two potentials. The thing that is measuring the voltage on the screen (and ultimately looking for the effect of your finger) has one connection to that screen, and the other to a reference that will almost certainly be the ground inside the phone. This "ground" is a common connection throughout the phone and probably to its metal parts. The idea is that when you hold the phone in your hand, there is capacitive coupling between your hand and the ground inside the phone. As to various conjectures oabout Earth ground, they may work because everything will ultimately couple capacitively to the phone to some extent. And as for "an ungrounded, capacitive object with sufficient surface area", well that is exactly what is shown in the illustration as a plate underneath the phone.
The charger or USB connection might well provide a ground connection into the phone, but that's not guaranteed. Opening the phone to get at it would work, but that would have limited practicality when it came to any end-user application.