They didn't.
The only "salting the fields" claim attributed to the Romans in the common knowledge is the supposed salting of Carthage at the end of the third Punic war. That's an invention of the 19th century, without support in the records of Polybius (sp?), Pliny the Elder or other historians of the Empire in the century or two following events. To the contrary, the fields of Carthage were siezed and given over to the commons for crop production, from which Rome recieved significant exports in the years following the razing of Carthage.
Some reading
A bit more (of the same)
More of the same
and yet more
Even wikipedia gets it right
There is a differing salting of the earth to be found in the Old Text's Book of Judges, which I do not read as a literal history. Abimelech (not a Roman) is said to have used salt after the fight with Schelem, though how much salt and how is open to some interpretation.
I may have nearly failed latin, but that doesn't mean I didn't spend many sleepless nights translating passages from the Histories. Part of the reason I took medieval history in college after Calc II kicked my . . . - less to read than in Ancient history.

Also greater application to the modern world.