A block of ice left undisturbed is in a state of thermal equilibrium. There is a portion of the ice that melts to form water and at the same time an equal amount of water solidifies to form ice. This establishes a state of equilibrium with the net amount of ice remaining constant unless there is a change in the thermodynamic conditions.
The addition of salt disturbs the equilibrium. Salt dissolves in the ice that melts to water. Water, with ice dissolved in it, has a lower point of freezing than pure water. Generally this applies to any solute that is added to water. All forms of dissolved impurities in pure water decrease its point of freezing.
This lowering of the freezing point does not allow an equal amount of water to freeze again as the amount of ice that melts to form water. Hence salt is able to melt ice.