The custom to wear disguises on Purim and even to appear as non-Jews is related to our forefather Yaakov’s wearing of Esav’s clothes when he received the blessings that were due him. It is as if we announce that just as Yaakov only had the outer appearance of Esav but was inwardly holy and pure, so are all appearances of evil in Israel only external, and inwardly we remain a holy people. At the time of Purim, Israel was likewise loyal to G‑d at heart; they only pretended to worship Nebuchadnetzar’s idol.

...hiding one’s face is proper on the day of Esther.

This custom has also been related to a verse in the Torah, as the rabbis comment: ‘Where does the Torah allude to Esther?’ It is said: ‘V’anochi haster astir panai…’ (‘And I will surely hide My face…’) (Deut. 31:17) ‘haster’ = ‘to hide’ – and ‘haster’ and ‘Esther’ are phonetically and grammatically similar). From this we learn that hiding one’s face is proper on the day of Esther.

Another reason: Amalek’s hatred to Israel came to him from his grandfather Esav, which was because of Yaakov’s wearing of Esav’s clothes upon entering to receive Yitzchak’s blessing – which Esav had regarded due him. Now we again disguise ourselves, to indicate that that disguise was not an unrighteous act, but that Ya’akov and his descendants justly receive the inheritance which Esav wanted.

[From Book of Our Heritage, vol.3, p.90]