Why do digital controllers need A/D and D/A converters?

Basically the "Plant" is the system being controlled. A good example is a boiler system. The inputs and outputs are continuous in time and amplitude. For example the boiler's burner may be controlled by the volume of air intake per second and the volume of fuel per second. These are analogue input quantities. Likewise, on the output side, the Temperature of the boiler is a continuous analogue quantity (say 0 degrees C to 200 degrees C).

That explains why the input and output of the plant are analogue.

The reason then, for the ADC and DAC are because the controller is implemented digitally. In a micro-controller for example, whose inputs and outputs are digital. Hence the need for the converters.

"Plant" presumably refers to the thing being controlled: for example a motor or a heater. The control system supplies power (continuously variable, analog) and the y(t) feedback signal is an analog feedback (such as motor speed or heater temperature).

The plant is the thing that you want to control.

It could be the temperature of a water tank, the speed of a motor, etc. The variable you want to measure is usually an analog value (not discreet) such as "a temperature". This value has to be taken into the digital world by an ADC (analog to digital converter) because your control system is digital and only understand digital numbers.

Then your algorithm will find a command for the actuator (for example: a heater for a tank temperature). This command is a digital number and it has to be translated into an analog value (the current for the heating element for instance), this is where the DAC (digital to analog converter) takes place.