Why are fae hurt by Iron?

They’re not.  The whole magic can be disrupted by a piece of metal is a myth the fae themselves have spread about to lure foolish humans into a false sense of security.

But, eh, in terms of actual folklore, unfortunately the people who first started telling these stories didn’t write much down so we’ll never really be sure what the thinking behind them was.  And it’s not just fae who were affected by iron, but ghosts, witches, demons – pretty much any magic or supernatural creature.  Iron in various shapes has often been used to ward off those things, or simply as a good luck charm, like a lucky horseshoe.

There was a story that Saint Dunstan, a blacksmith, once nailed a horseshoe to the devil’s hoof, so that may be where that came from (although there are possibilities as well).  As far as fae and other creatures, it may just be to do with the superiority of Iron tools and weapons over, say, stone and bronze.  One theory I’ve seen mentioned is that the Ancient Britons, for example, displaced older peoples, some of whom may have tried to wage a kind of guerilla war against the invaders as they were no match for the iron weapon but knew the lay of the land better, and that may be the origin of at least some elf and fairy myths.  But that’s just conjecture really – we’ll never really know.