Ice is not at 0 degrees Celsius, it is below that. By adding the salt, the freezing point of water is lowered further so that the entire liquid mixture itself can become lower than 0 degrees.
Water cannot drop below its freezing point until all of it is frozen. By lowering the freezing point, you let lose more heat before it gets stuck at one temperature. This is called the heat of solidification.
The same holds true for boiling: the water will not exceed 100 degrees Celsius until the entire amount has changed to gas. This is called the Heat of Vaporization.
Adding any substance to water (or any liquid, for that matter) that creates a solution, will both lower the freezing point and increase the boiling point. This is because the salt throws off the equilibrium of the reaction, and also affects water’s ability to form a crystalline solid structure.