Freezing point depression is the phenomenon where adding a solute to a liquid causes the freezing point to drop. While this may seem simple, it's not. Actually it is, it's just not intuitive. While the concept is pretty simple when you are dealing with water you are trying to freeze, it becomes a little stranger when you talk about adding salt to a mixed phase system of ice and water.
The most common such system is the ice water bath around an ice cream maker. You start with ice and add salt. Surprisingly, as the ice melts, the temperature of the ice and the water both drop without removing any heat from the system.
In simple terms, at 0 C, ice and water are constantly melting and freezing, just in the same amount so there is an equilibrium. If you add salt, the freezing point of the water drops but the melting point of the ice stays the same. So ice still melts but water stops freezing. As the ice melts, it absorbs heat. The interface between the ice and water will cool down first, followed by the rest of the water and ice, until the system reaches the new equilibrium temperature of -20 C (or whatever the freezing point is depressed to depending on the amount of salt added).
Here are the numbers that govern the cooling of ice and salt water:
Specific heat of water: 1 cal (4.186 J) per g K
specific heat of ice: .51 cal (2.135 J) per g K
Heat of Fusion of water: 334 J per g
To melt 1 kg of ice requires 334 kJ.
To cool 1 kg of ice from 0 C to -20 C requires 42.7 kJ.
To cool 1 kg of water from 0 C to -20 C requires 83.72 kJ.
The solubility of salt in water at 30 C is 37 g/100 mL or 37 g/100 g. At 0 C, it will be a bit lower than this. When salt dissolves in water, it absorbs about 51 J/g. So if you dissolve 370 g of salt in the kg of water, you will absorb a total of 19 kJ.
The heat capacity of water is 4.2 J, so 19 kJ will lower the temperature of 1 kg of water by about 4.5 C to -4.5 C. Since you've lowered the freezing point of the water, none will freeze.