Exioxx
You are correct that there is a link between systems for automatic switching to occur; this can either be via data bus links between computers or simply by relay connections.
Trying to keep it simple;
Depending on the system; computers on different power supplies will be connected through data buses, each computer comparing and monitoring against the computer in control to detect a failure and then force a transfer to a good computer.
Relays often provide discrete inputs to computers as to status of power and other circuits. A failure of power on a circuit can be the trigger to initiate the system on standby.