One of the dangerous byproducts of nuclear fission, and a component of fallout from a nuclear explosion, is a radioactive isotope of iodine. The iodine in your body tends to accumulate in your thyroid, where the radioactive kind could cause cancer. If you are exposed to the radioactive form of iodine and eat an excess of normal iodine, though, it will compete with the radioactive kind and cause less of it to accumulate in your thyroid and presumably reduce your risk of cancer. Seaweed is a good source of iodine, which is why it might be recommended for protection against radioactivity. Obviously, there are other sources of radioactivity besides iodine, so this might be of limited benefit.
Bob Scene is correct. That article sort of implies that seaweed actually protects you from radiation; this is false. It just prevents any radioactive iodine you might happen to ingest from concentrating in your thyroid.
Also, there is no longer such thing as the Atomic Energy Commission. It was replaced by the Nuclear Regulatory Commission decades ago. In any case I doubt the AEC/NRC have ever seriously recommended that people eat seaweed every week. In the event of a major release of fission products, you can take potassium iodide pills (which, AFAIK, the government stockpiles for this purpose) and get pretty much immediate protection against radioactive iodine. In other words, there’s really no advantage to eating lots of iodine-rich food in preparation for disaster.