My understanding is that iodine is extremely rare in the earth's crust, and so it is naturally found only in some vegetables close to the sea coasts. Given, that humans have lived away from sea coasts for a very long time, how did they make up for iodine in their diets?
Iodized salt was started in the United States in 1924. Before refrigerated transport, many parts of the United States suffered from goiter--a disorder of the thyroid gland due to lack of iodine.
You can naturally get iodine in your diet by eating seafood and vegetables from iodine-rich soil. The areas that did not have access to seafood and iodine-rich soils had a disproportionate number of goiter cases; so much so that it was termed the goiter belt. EDIT: There's a map of goiter prevalence for 1920 in this pdf).
Iodized salt was introduced to help combat goiter.
Why salt? Because it was a relatively cheap substance that everyone used on their food. Plus, the iodide salts used taste similar to table salt.