From what I hear this problem happens more in Europe than in the USA. I have no idea if this is because of the 110/220V difference, or because of the type of breaker/electrical wiring used.

This problem is caused because when you switch on the game, the transformer (and other parts of the game) suddenly require a lot of electricity. It causes a sudden load, the main breaker of your house thinks there's a short somewhere in the circuit and resets. Bam - your whole house is without electricity..
There's also a reason why it doesn't happen all the time. AC is a wave which goes up and down, 50 (or 60, depending on what part of the world you're in) times a second. When you switch the game on while the wave is at the bottom, you usually have enough margin so the breaker doesn't act, switch on the pinball machine while the wave is at the top and the circuit gets overloaded.

There are two solutions for this. The first is a workaround - you can add more resistance to the wiring (so the main breaker becomes slower to react), try to plug in your pinball machine in a long extension cord instead of directly into the wall. The extension cord can help solve your problem.

The correct way to solve the problem is to install an NTC. NTC stands for Negative Temperature Coefficient, another word for it is thermistor. It's a resistor that changes its value with its temperature. It's job is to limit the inrush of current when you turn on the machine.
Turn on the pinball machine and the thermistor will limit the inrush of current because of its resistance. Once the machine is on, the resistance of the NTC lowers, not influencing the circuit anymore.

Source: http://www.flippers.be/basics/101_sys11_house_fuse_trips.html

Yep! The NTC resistor (thermistor) did the trick!
Hopefully this is helpful for other people in the future as well.