Correct execution of string damping technique is a very important requirement for achieving a polished performance.
The main reason for damping a note, apart from the obvious one of artistic articulation, is to prevent a sustained open string from clouding the harmony when there is a harmonic change in the composition. For example a low open A bass note will continue to sound for as long as the string will sustain. If during this sustain the harmony changes to a dominant E chord the open A string will combine with the newly played notes to produce an unpleasant discord. In this case the open A string should be damped slightly before the change in harmony. This is achieived by placing your right thumb momentarily back on the vibrating A string just before playing the next chord.
The other method of damping is to simply lift the left hand fingers from the note but obviously this will not work on a open string.