Nice question.
Although pure water freezes at 0°C (32°F), water that has salt dissolved in it has to be colder before it freezes. If the water has as much salt dissolved in it as it can hold (that's called a saturated solution of salt), so that any further salt would just come out as crystals, the freezing temperature is around -21 °C, or about -6 °F. If your freezer isn't colder than that, the part of the ice touching the salt will start to melt. If you've put so much salt on the ice that the water can all melt and form a saturated solution, and still leave some salt crystals, then it will all melt. If you've put only a little salt on, it will melt some ice until the salt crystals are gone. Now as more ice melts the solution becomes less salty, more like pure water. So its freezing temperature goes up. At some point its freezing temperature will be the same as the freezer temperature, so the freezing will stop. You'll have some ice left, and some salty water.
What is interesting is that this effect is used all over the place. Often, salt is put on roads to melt ice. If there's a lot of ice, you need a lot of salt. If the temperature drops below -21°C, it won't work at all.
You also wanted to know why it works, why saltwater has to get colder than pure water before it freezes. We've got some other answers on that, which you can find by searching this site for "saltwater". Briefly, the ice is a crystal, an almost perfect array of pure water molecules with almost no salt in it. To make that out of pure water requires limiting the ways the water molecules move around. To make that out of salt water requires BOTH limiting the ways the water molecules move around AND limiting the ways the salt can move around (it's stuck in the liquid, or in separate crystals), which is harder to do.
Adam and Mike W.
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