DIAGNOSIS That's mineral scale—calcium and magnesium from your hard tap water. It's harmless but unsightly. It forms on the wet plastic walls of the reservoir and then flakes off into the water.
FIX You need to descale the reservoir. 1) Make a cleaning solution of one part white vinegar to three parts warm water. 2) Fill the reservoir with this solution and let it soak for 15-20 minutes. 3) Scrub the inside with a soft bottle brush or a non-abrasive sponge. 4) Rinse it thoroughly with cold water several times until the vinegar smell is gone. Do not put it in the dishwasher.
IF THAT DIDN'T WORK The flakes might actually be from a deteriorating filter if it's well past its replacement date. If descaling doesn't solve it, put in a new filter and see if the issue clears up.
HEADS UP To prevent this, rinse the empty reservoir with cold water and wipe it dry with a soft cloth every time you change the filter. This removes the mineral-laden water film before it can dry and crystallize.