Someone who is from the small town of Ascoli, Italy.
Original word is DiAscoli. 'Di' in Italian translates to 'of' and Ascoli being the town. Thus DiAscoli meaning 'Of Ascoli'.
When the family of DiAscoli's migrated to the United States they were forced to Americanize their name. They dropped the 'i' and added an apostrophe (which was common)