MOVs can be found with almost any voltage rating nowadays. But for AC circuits in the US, a common value is 130V RMS working voltage. What this means is that when the peak input voltage goes above \$130 * 1.414 = 183.8V\$ then the MOV will start to conduct (like a Zener diode in a DC circuit would). As pointed out in other answers, the facility (house?) wiring provides some small impedance that allows the voltage to drop across, thus limiting the voltage at the MOV node.
As the input voltage increases more, then the MOV will conduct/turn on more. They say "more grains will conduct". As more grains turn on, each one being like a tiny zener diode, the device will conduct more current and dissipate more power as heat.
This can go on for a very long time, as long as the MOV can cool down between transient events.
Failure Mode
During a common over voltage event (like 220VAC appearing on the 120VAC lines) as the MOV turns on, the heat tends to be highly localized inside the device, and will create hot spots. These hot spots will eventually fail and instead of a low impedance, those spots are now high impedance, but still conducting. This creates more heat, and now a chain reaction occurs and the device catches fire in a spectacular fashion!
The fuse in your schematic is usually a "thermal" fuse. Meaning that is is placed right next to, in contact with, the MOV, and if the MOV conducts too much current, then the fuse opens like a normal fuse would. But also, if the MOV gets too hot it'll also open in order to prevent the MOV from catching fire.