We use it to melt ice in both cases.

Salt lowers the freezing point of water. Fresh water freezes at 0°C. With enough salt, you can lower that quite a bit. At 10% salt concentration, you're down to about -6°C, and at 20%, the water freezes at about -16°C.

So, suppose you fill a bucket with ice that is -12°C, and you add 20% salt. This lowers the freezing point to -16°C, but the ice is warmer than that, so it melts. And the action of melting actually absorbs quite a bit of energy, so this will lower the temperature. So you get a bucket full of ice water (i.e. a mixture of liquid water and chunks of ice) at a temperature of -16°C. Stick a metal bowl in there and you can make ice cream in it.

Why do you want this and not just ice? Well, solid ice cubes or shavings don't make good contact with the bowl that contains the ice cream ingredients, so the heat doesn't transfer very quickly from the bowl to the ice. Liquid water, on the other hand, makes great contact with the bowl, but normal liquid water isn't cold enough to freeze the ingredients. So the perfect combination is liquid water that is also below the freezing point of your ice cream ingredients.

Note that we added salt to make the ice in the bucket melt. This is no different from putting salt on roads to melt snow or ice.