Italian Jews in the middle ages were the first to adopt the custom of dressing in costume to celebrate Purim. The Italian Jews were inspired to wear costumes by the Roman Carnival, naturally. But the custom spread and stuck, likely because it is so appropriate for Purim.

Hiding our identity by dressing in costume is a way for Jews to experience the Purim story. The story is chock full of people mis-representing themselves and concealing their true identity. Esther is the major incognito who conceals her Jewishness from the King and becomes Queen. Later she reveals she is a Jewess to save her people from Haman's plot to kill the Jews. Other cases of mistaken identity include:

• Mordecai (Esther's Uncle and informant) hiding his language abilities and thus eavesdropping on the plot for Jewish extermination;
• Haman (the King's right hand man and striving Jew killer) is mistaken for Mordechai and thus, as it's discussed in the Talmud, Haman's daughter dumps her chamberpot on top of her own father's head! Oops.

Shakespeare must have gotten his inspiration from the Purim Megillah.

Read more about Purim on Wikipedia's Purim