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.