So what is the ground fault thing? How does that work?
Normal breakers and fuses pop when the total current exceeds the limit for long enough. This will keep your wires from burning up, but says nothing about where the current is flowing.
GFCI breakers compare the outgoing current against the incoming current, and pop if there is even a very tiny imbalance (milliamps) between them. This is a definite sign that some of the current is flowing somewhere it isn't supposed to, and making its return path back to the supply through some other means. This might be another piece of equipment. It might be you.
The "green wire" (aka Equipment Grounding Conductor or EGC) is intended to intercept those stray currents, and give them a low-resistance path away. By grounding the chassis and enclosure of powered equipment, it becomes much more likely
that any stray current will be intercepted by something safe, rather than passing through your hands, feet, and in between, your heart muscles. Suppose the live wire inside your old-school toaster shorts to the metal case. If the case is grounded, you will not take very much current if you put your hand on the toaster. If the case is not grounded, you will basically be putting your hand on the live wire.
A short from a live wire to ground is called a "ground fault", which a GFCI breaker will detect instantly. But GFCI breakers don't depend on EGC. They measure imbalances in supply and return currents. A lot of corded power tools are "double insulated" and they didn't require EGC green wires. GFCI breakers would work just fine with these tools, detecting if a short somehow got through the "double" layers of insulation.
I know you need them near the sink and in the bathroom, but why?
Because wet environments are where leakage currents are potentially the most lethal to human beings. Wet skin is a much better conductor than dry skin, so the human body will pass more electric current.