Let's take water as our analogy. Water flows to meet demand in the form of open taps. But very few of those taps are strictly regulating, and the outflow is a function of how far the tap is opened and the pressure in the system. Put more water into the mains and the pressure goes up, therefore more water is delivered at the tap. If your house has pressure regulating valves, you won't see this, but the pressure is then further increased at someone else's house.
That analogy doesn't really work very well, for two reasons:
1. Water pressure is more closely equivalent to voltage, not amperage. Adding more solar panels increases the amperage, not the voltage.
2. Most electrical equipment is strictly regulating (ignoring inrush). Resistive loads consume a consistent amount of current regardless of how much current is available. That's why it doesn't matter whether you power a 12V bulb with eight AA batteries or a 12V car battery. The latter can provide a lot more current, but the bulb still draws just as much current as it needs.
I think a better analogy is to think of the voltage as the height of a water tower, and the amperage as its diameter. If you have a ten-foot-diameter tower that forms a 50-foot column of water, the pressure is proportional to the 50-foot height of the water column. An overheating condition would be equivalent to the pipe breaking because someone is sucking water out of the pipe faster than the pipe can pass it.
If you expand the tower to be thirty feet in diameter, the column is still about 50 feet high, so the pressure is about the same (assuming the sides of the tank are vertical and the bottom is flat). However, doing so allows you to add more pipes and/or larger pipes out the bottom so you can provide water to more houses without drawing down the reservoir too quickly (and thus causing... what, a vacuum in the water tower? This is where the analogy starts to break down unless you're talking about a battery).