A Red Letter Day is noted in the dictionary as a noun that describes any day, "as pleasantly noteworthy or memorable".
The idea behind the line was that each piece played a part, or told a story in someones' having of a red-letter day. So, in this case the Ugly Duckling is having a Red Letter Day, because he knows not, and cares not about his awkwardness. He is content, and no amount of perfection can change his sweet carefree attitude. He is having a Red Letter Day. Everyday.