It may help to think of this like a car battery. With a car battery, you have a + terminal (positive) and - terminal (negative). The wire leading from the negative battery terminal is often run straight to the car's frame, which being made of metal, conducts electricity. That way, any hot wire (red) that touches the car's frame is then connected back to the battery. 
Actually, the correct way to jump start a car with battery cables is to hook the RED cable to the battery (and do that first), then hook the black cable to the car's frame somewhere away from the battery. That way, when it sparks, the spark is not anywhere near the battery. In older days, prior to sealed car batteries, the gas given off from a charged battery was hydrogen, which is flammable.....read "will blow up in your face"....literally..........which is why you wanted the spark to occur far away from the battery.
So with that car's system, anything electrical is powered from the + battery by a network of red wires. Red wires are HOT wires. They are insulated and the metal wires inside them can never touch anything that is grounded. If they do, current will flow and the that wire will short out. But connect that insulated red (hot) wire to a light bulb.......say the little light that comes on when you open the trunk. But it doesn't connect directly to the light. It goes through a switch first. When you open the trunk, the switch is released, allowing the current to flow.......to the light bulb, which is grounded to the frame. So the current flows down the red wire, through the switch, through the light bulb's filament back the frame and thus  back to the battery. What you see is the light comes on. Close the trunk, the switch is cut off, current stops flowing and the light goes off. 
Your fencer is doing the exact same thing. You have the red wires (this is the HOT one) and probably a green wire (this is the same as black, ground or negative). The HOT fence wire can never touch anything grounded or the current will flow (fence will short out and take away the potent shock potential). So the hot wire is what you place in the path of the animal. You want it fully insulated from any pathway back to ground, but fully exposed in such a way the animal (the switch) is going to touch it to create a pathway back to ground. 
The grounding system (think of it as a system) should be as large as it can be. Indeed, the entire surface of the earth and anything that conducts electricity that can be attached to it is part of your grounded system. But to be grounded, it does need to be attached to this entire ground system. Your ground rod is attached......the black metal fence (I assume it is metal and at least the posts are buried in the ground) is grounded. Hardware cloth clipped to metal fence posts buried in the ground is grounded. Hardware cloth screwed to PVC posts is insulated from the ground and is NOT grounded, unless you connect a grounded wire to it. Then it is. If you hung the hardware cloth on PVC posts, I would absolutely ground it, so it becomes attached to the larger ground field. No need to run a bunch of extra ground wires all around unless there is nothing there that can be grounded if you don't.
For example, if you had a PVC or board privacy fence, and wanted to run a hot wire along the top......the PVC or board privacy fences are not conductor's of electricity. So an animal clinging to those is not grounded and can touch the hot wire with immunity (impunity?). So in that case you may actually want to run or install a ground wire alongside the hot wire to ensure the animal will be grounded when it touches the hot wire. It has to be done in such a way to touch both. Like finding a way to have the animal stand on or cling to the ground when it reaches out to grab the hot. Or run them close together so it tries to crawl through the two of them, touching both at the same time. 
Hope this helps!