Why would bridging a varistor with the multimeter leads on any setting have caused the unit to turn on?
A varistor behaves electrically as a resistor. Your overvoltage event undoubtedly burned out the varistor so it appeared as an open circuit. This prevented the AC from turning on. When you bridged the varistor with your multimeter, you inserted some resistance in the circuit which was enough to allow the AC to turn on. Since you don't know what setting you used it is difficult to give more details. Note, however, that a multimeter will appear as a resistance to the circuit being measured for almost all of its settings whether current, voltage or resistance.  The exact value of resistance will vary with the settings and can range from an ohm or two up to many megohms. Again, without knowing what you did, no more can be said. As to not being able to reproduce the original scenario, you probably haven't tried the original multimeter setting (since you are blowing fuses, I assume you have not run through all of the possible multimeter settings).