For example, how do you explain why the perimeter of this staircase does not converge to $\sqrt2$? Or, why isn't $\sqrt{(-1)(-1)}=\sqrt{-1}\sqrt{-1}$?
I would say, the reason is simply because they cannot be proved. But non-mathematicians don't find such explanation satisfactory. They seem to be more satisfied by sophisticated explanation. For the staircase paradox, my friend who studies physics reasoned that it's because the perimeter is of dimension 1 not 2. It makes no sense to me, but they found such explanation more sensible.