The spots you're seeing here are actually mineral deposits from your water. This is fairly common in areas that have hard water—calcium and magnesium in particular will leave deposits like these.
Luckily, getting rid of these deposits is very easy: Just wash them with vinegar, which will dissolve the minerals. Gentle abrasion with a scrub brush will help!
You could also do this with your dishwasher by adding some vinegar (or any of the hard-water-management additives on the market) to the wash.