Interfacing is the process of connecting a microprocessor to the rest of the circuit or to external devices. Even in the simplest of computer systems, there is some input device like a keyboard. So how does the microprocessor know that we have pressed a key? When we send text to a printer, how does the printer tell us that it is ready for more input?
In a general purpose microprocessor-based system, if it is to do anything useful, there must be inputs and outputs. The external devices must therefore communicate with the microprocessor. In some cases, the microprocessor takes the matter into its own hands and sends data out as part of its program but even in this case it normally allows the external device to help.
If a microprocessor-based system were used to heat some water, it is easy enough to imagine the program switching on the power supply and sitting there doing nothing for 10 minutes. It would be a better use of the microprocessor to leave the heater running and wait for a thermostat to signal that the water has reached the required temperature. This thermostat signal would arrive at an interrupt pin on the microprocessor.