When the string is plucked harder, the force applied to it is greater, causing the string to vibrate with a larger amplitude. This larger amplitude results in a louder sound.
The relationship between the force applied to the string and the amplitude of the vibration is governed by Hooke's law, which states that the force required to stretch or compress a spring is directly proportional to the displacement of the spring from its equilibrium position. In the case of a guitar string, the string acts like a spring, and the force applied to it causes it to stretch and vibrate.
In addition to the force applied to the string, the loudness of the sound also depends on other factors, such as the tension of the string, the thickness of the string, and the distance from the bridge to the sound hole. However, for a given guitar string, the harder it is plucked, the louder the sound will be.