Dear <Name@Withheld>,

The earliest source I've found mentioning the custom to dress up on Purim is the responsa of Mahari Mintz (late 15th century). He discusses the issue of whether men may wear women's clothing as a costume, and vice versa. Obviously, the custom to wear costumes was well established by that time.

Among other things, masks on Purim symbolize the fact that the whole Purim incident was wrapped in "the hidden nature of G-d's countenance." The Purim events all happened in a seemingly natural manner. G-d's name isn't even mentioned in the Book of Esther! The very word "Esther" means "hidden," and it appears in the Torah in the Hebrew phrase "I will hide My Face." (Deuteronomy 31:18)