"It is known that Iodine deficiency in adolescents results in enlargement of the testes, just as it results in enlargement of the thyroid gland without concomitant production of androgenic hormones and virilization."
Iodine insufficiency appears to cause swelling in many different organs, so it shouldn't be surprising if prostate enlargement is also a result of a shortage of iodine. The mode of action of compositions known to reduce prostatic enlargement may be by a mechanism which perturbs and causes a re-distribution of existing bodily Iodine to the prostate.