The idea is that salt will poison fields, making them impossible to farm and therefore making a city on that spot unlivable.

This is much more metaphorical than literal - it should be noted that no one suggested Rome literally plowed over the fields of Carthage until Barthold G. Niebuhr about 2,000 years later. This myth was compounded around 1930 by B.L. Hallward, who added the salting part under the influence of Biblical imagery.

The origin of this comes from a ritual which began in the ancient Near East and spread into the Mediterranean and India, where the priests of a conquering army would scatter a few handfuls of salt on land consecrated to the gods of the defeated. It's possible (although nowhere documented) that Roman priests performed such a ritual to expel the Carthaginian gods from the defeated city.

Now that said, salt was not extremely expensive at many places at many points in history, and especially not in the Roman Empire, which allowed trade between three continents centered on a vast sea. Peat salt was easy to come by in northern Europe, slaves produced salt by the ton from mines in Arabia and Africa, and every coastal village had brine ponds to harvest salt from. A day's wages for an average laborer would buy a half-gallon of salt (two liters), enough to last a family for a good while.