I think, and hopefully someone more knowledgeable will show up to correct me, that Iodine is fat soluble and that it "dissolves" into fats where the double bonds are in the carbon chain. It isn't that the Iodine is destroying the fat molecules, or making them saturated; it's that the Iodine has an attraction to the sites of the double bonds in the carbon chain. The more double bonds in the chains, the more places there are for iodine to find a niche to hide in, so to speak.