When the mass ‘m’ is hanging from a string and is in equilibrium without breaking, then the tension in string is equal to

weight of the mass. But when the same mass is set onto oscillation it oscillates in an arc of circle, in a circular motion with the length of the as the radius of circular path. Due to this centripetal acceleration comes into picture thus, centripetal force acts on the mass in addition to the tension in equilibrium position. So, the string breaks due to the oscillation.