Old computers wouldn't turn themselves off, they had no mechanism to control whether they remained on. Power was controlled by a heavy duty switch on the side of the PC (some manufacturers moved it to the front or something too, but many had it on the side/back).

When ATX became a thing, power controls were done by a trigger wire from the main board to tell the PSU to turn on fully. This is how things are still done. With 80+ Silver/gold/whatever rated PSUs they actually don't really turn off anymore, power draw just drops to next to nothing when the system is "off".

The hardware switch would physically disconnect the power to the PSU. So when you shut down, this message was displayed, most notably by Windows 9x, to inform you that it had finished the shutdown process and you could flick the switch to turn the power off, and it wouldn't cause any damage to the system.