Why does salt melt ice?

Short answer: salt melts ice by lowering the freezing point of water. Even an ice surface near 0 °C has molecules continually exchanging between solid and liquid. Dissolved salt makes the liquid phase more stable at temperatures where pure water would freeze.

The important ingredient is brine

Once salt contacts a little liquid water, sodium and chloride ions spread through it. That solution has a lower freezing point than pure water. At a temperature such as −3 °C, pure water favors ice, while sufficiently salty water can remain liquid, so more ice melts to reach a new equilibrium.

Why the mixture can get colder

Melting ice requires energy. In an ice-salt mixture, melting can draw heat from the mixture and its surroundings while the depressed freezing point allows liquid water to persist. Chemistry demonstrations use this effect to drive an ice-water bath well below 0 °C.

Salt has limits

Ordinary sodium chloride is not equally effective at every temperature. There is a practical low-temperature limit, and the exact behavior depends on concentration and conditions. More salt also stops helping once the solution approaches its relevant phase limits.