There's a chemical potential (i.e. a driving force) for salt to dissolve in water, since there is a higher concentration of sodium and chloride in the salt crystals than in the water. By the same process, salt attracts water to its surface (i.e. it deliquesces). These two processes get the dissolution started, resulting in a film of brine at the ice/salt interface. From there on in, it's a simultaneous heat and mass transfer process looking for an equilibrium.
As the salt dissolves, the resulting brine has a depressed freezing temperature relative to pure water. The "heat of solution" has to come from somewhere- in an insulated container, the resulting mixture temperature drops to compensate for the heat of solution, such that 33 parts of salt added to 100 parts of ice will produce a brine/slush mixture at ~ -21 C. (that's how my grandfather's generation made their ice cream, before mechanical refrigeration made it easier). In an open system, the heat of solution will be withdrawn from the surroundings and the equilibrium will be shifted away from salt/ice and toward brine until one or the other is gone.
Salt is useless below temperatures which will freeze the resulting brine mixtures. The colder you get, the more salt you need- and beyond ~ -20 C it becomes practically pointless to apply salt. At those temperatures, we generally switch to sand as a cheaper and lower environmental impact option to give more traction on ice.
On a busy road, you have hundreds of vehicle tires doing work on the snow, and vigorously admixing the snow/slush with any salt present. It's the heat generated by friction, plus the mobility of the brine/slush to flow and via spray, that accelerates removal of the snow from the road. Hence it gets all over every surface of your car, and also into streams and soils nearby....