I believe there are many things going on.

The skin is the largest organ in the body and needs:

protein to build and repair cells
water to hydrate and carry off impurities
fat to plump up and keep cells flexible
vitamins to prevent aging and heal itself.

Sugar is not good for cells, it binds with cholesterol and scars cell walls making them rigid and inefficient at absorbing nutrients. Further high insulin levels speed cell aging.

A diet high in grains can cause allergic reactions which irritate the body.

Impurities are secreted through the skin and through urine. If it does not get enough water for sufficient elimination, the only path left is through the skin.

Low carbing provides sufficient protein, water, fat and, with supplements, vitamins along with the avoidance of raised insulin levels to promote healthy skin.

Didn't mean to sound like a lecturer!