I've had it explained to me many times but never completely understood it. What I do know is that when working on equipment with live chassis (such as old monitors, TV's and radios), it greatly reduces the risk of shock.
Many monitors require one, and will be damaged if powered up directly from the mains.
120VAC is distributed via a single super high voltage line from the power plant to your home and then stepped down via a transformer near your home. Since this is only a single wire, the "return" of the electricity to form a circuit is actually done through the earth itself.
Outside your home, the hot wire comes from the step down transformer and the neutral wire comes from a stake in the ground by the pole. Once in your home, another "ground" is driven into the ground giving you the third wire so that when something shorts it finds a path to ground sooner than all the way back at the transformer.
Now, you plug anything into an outlet, you have 120VAC coming in over the hot wire coming from the plant, going through the device, and back out the neutral line which makes it back to the ground next to the power pole.
Now, if at any point you were to touch the hot wire, your body will act as the conductor driving the voltage directly to ground through you instead of through the neutral wire. Bang, you're dead.
However, if you isolate that power, so that EARTH GROUND has nothing to do with the power, then you provide the safety you need to not kill yourself touching the power source. Just like a car battery -- touch the +12V line with one hand, and have your feet in mud --- you won't get a shock because the two have no connection between each other. However, run a wire from the battery's - terminal into the same mud, then of couse you will get a shock (or simply hold one hand on - and one hand on + completing the circuit that way).
So, an isolation transformer does just that ... it provides a "battery like" secondary power source off the primary power source. Therefore, you can safely touch EITHER side of the power feed coming off a isolation transformer and NOT get a shock since the power has no desire to head to ground since it's not connected to ground. Touch both and sure, you're still dead.
Now, as to why this is important for monitors: Older TV's and monitors are designed so that their "negative/ground" side is directly connected to one of the incoming power leads. Without isolation, the chassis is hot (meaning the negative traces, metal heat sinks, and even the frame and ground strap around the tube). Connecting it to the power line directly ensures first that the entire thing (in areas you THINK are safe) are actually HOT as they are carrying the flow of electrons that WILL want to find the quickest path to ground -- through you. Also, most monitors actually will have their bridge rectifiers and other components fried when connected without a transformer.. perhaps a safety design thought to ensure they aren't left hot.
Open up a consumer TV from the 80's and plug it in and then meter against something metal inside and you'll find it is indeed hot and can kill you if you touch it. Thats why technicians plugged them into iso's to work on them just like we plug arcade monitors into iso's. The difference is that on TV's of that era, the only external connections (with the case closed) is the antenna inputs, and those were protected by a large disc capacitor that prevented voltage from leaking out while allowing the antenna frequencies in. Newer TV's used opto-isolators to isolate the video inputs and other ports until they switched to onboard switch mode power supplies and were no longer made hot.
If you actually found a monitor that didn't fry immediately on plugging in directly, and then connected your game board to it, you'd find you'd send 120 volts directly back through the video cable and blow up your circuit board.
Newer monitors don't actually need separate iso's since they in fact have them already on the chassis. They're tiny and work in the same fashion switch mode transformers work on PC's and the black boxes in your arcade cab. They increase the frequency of the power thus reducing the size of transformer needed and then decrease it on the other side.
You can NEVER hurt anything by putting an iso on something that already has one built in, or running two big iso's one after another, but you can hurt your monitor, gameboard and yourself if you plug in without one for one that needs one.