The linked article discusses specific details of controllers manual to automatic switching and is obviously not intended to teach basic terms of control engineering. Nevertheless it gives you an idea about the purpose of mode switching, if you just read it thoroughly.
Consider that a controller is not necessarily always active. It may be activated and deactivated at times. Manual mode means that a controller is deactivated, in other words not reacting on set point and process value variations. Instead the controller output ("manipulated value") is either hold at a defined initial value or controlled by a different source. "Bumpless" is a keyword in mode switching, start controller operation without any unwanted excursions of the manipulated value.