The way that salts melts ice and snow is quite simple. In plain words, when there is the presence of salt near ice or snow, the ice starts melting at temperature below 0 degree Celsius. Usually without the presence of salt, the ice will melt only if temperature rises above 0 degree Celsius. This means that ice cannot stay in solid state and will turn into liquid state due to the presence of salt mixing with the thin layer of water on ice. People have discovered this property of salt and ever since had been using salt in large scale to clear snow and ice in cities around the world.
Salt also offers more grip and friction for the car tires with the road, to grip on during snowstorm or freezing rain. Salt is much cheaper than other products because it is readily and widely available in the market. Contrary to much belief, salt doesn't cause much damage to grasses or green lawns. The grasses had evolved to tolerate the presence of salts and in some cases benefits from the absorption of salts. Tree roots had also evolved to tolerate salts and it too does get some benefits from absorbing the salts.
No doubt that the salt causes damage to our roads and shorten the lifespan of our roads, but in the long run, the savings government get from using cheap salts will offset the cost of having the repair the roads afterwards. There are other alternatives mixtures of salt and other additives but they are generally more expensive and cost more in the long run. In the end it is all about saving money for the government and at the same time providing essential services of clearing streets and roads.