If you rub at your eyes, you'll get dark circles under them.
Rubbing at your eyes will darken the bag under them.
「痒いからと目を擦ると色素沈着で黒くなるよ」は英語で「If your eyes are itchy and you rub at them, the pigmentation around your eyes will darken.」という表現が良いと思います。
痒いーItchy
擦るーTo rub/To scratch
色素沈着ーPigmentation
黒くなるーTo become black/To darken
アメリカ合衆国 
