Partially it was because potassium iodide (a form of iodine that is convenient) is also a salt, like sodium chloride (table salt). So "put the salt in the salt" is pretty obvious.

It was also wildly successful because at the time, salt was turning from an expensive good of varying quality produced by thousands of companies to an extremely cheap good of uniform quality produced by a near monopoly in the United States. Public health advocates were able to get just a couple of companies on board and iodize basically all the salt, which then reached basically all the people, basically solving the problem completely overnight.