Effectively, the salt molecules stop some of the water molecules from touching the part of the water that is becoming solid. At 32 degrees Fahrenheit, water molecules enter and leave the freezing state at an equal rate.
When salt is present, however, it upsets this process, as it prevents the molecules from coming together. It is like having a handful of weak magnets. They come together easily if nothing is in between them, but if other things are present, such as marbles, the magnets have a more difficult time attracting one another.
Water with salt in it has to be colder for it to solidify. Lowering the temperature of water makes the molecules move more slowly. In time, the molecules entering and leaving the solid part of the water equal out again, as everything is moving more slowly, and it is able to freeze.