Good answers but I still miss the most essential reason:
The game PCB have a crystal on them. This is a certain kind of crystal which has one very nice habit: when put under an electrical charge, they start to resonate (shake if you want) at one extremely precise frequency.
F.i. this frequency can be 6 MHz which means it "vibrates" 6 million times per second.
This is usually too fast for late 70s/early 80s stuff so the frequency gets divided back to a lower speed and then is fed to the micorprocessor.
Now, the voltage put on the crystal will always be (more or less) the same and it really doesn't have to be all that stable.....the crystal will do its work on the same frequency and hence the game will play exactly the same all across the world.
Interesting side-note: the monitors produce the pictures at (about) 60Hz. This is not caused by the mains but is also controlled by the electronics on the PCB. This is why the arcade screens have a relatively steady picture. This is accepted as normal these days, but we in Europe were used to 50Hz TV pictures, which really do flicker quite a bit of you are used to higher frequencies. Following this, I was always amazed at the stability of arcade game pictures (and we weren't used to anything that could produce RGB pictures too.)
Now.....there is one are where 50/60Hz does still matter and that's with transformers. Most of the main transformers can accept bothe frequencies, but f.i. the ballasts for the marquee lighting are really designed for 60Hz......they work at 50, but it's not an ideal situation for them.....but hey we don't have them powered up 24/7.....