In the United States the military has been a source of American slang since the Revolutionary War and much of it has become increasingly embedded in popular culture. To chew means, "to bite and grind with the teeth; to masticate, as food, to prepare it for deglutition and digestion," so it's not too big of a leap to connect the idea of biting into something with a grinding reproach. Alternative phrases for the transitive verb can be dress down, take to task, or rake over the coals.
No doubt the phase came from across the pond because a good chewing out is hardly a new phenomenon. In Europe people were being called on the carpet since the beginning of the 19th century when a servant was said to "walk the carpet" when they were summoned before the master of the home for a tongue lashing. Even earlier than this, people were being read the Riot Act. In 1715 English magistrates were responsible for the control of unruly citizens. "If more than twelve people didn't disperse after the Riot Act was read to them the magistrate could order their arrest"and by the 1930's judges were said to be throwing the book at disorderly citizens for breaking the law. The 1986 Public Order Act made the Riot Act obsolete, but there is some evidence to suggest that the phrase describing a verbal reprimand may have originated even earlier because the OED offers a citation from as early as 1230 for chewed used to illustrate "to worry with reproaches." Since then the formal censure or reproach of a subordinate by a superior has escalated from getting an attitude adjustment to a good bawling out.
Some say that the military may have begun bandying the phrase about as early as World War I. One etymologist cites a passage, "H.Q. is chewing out my arse why we're not flying right now," from Stephan Longstreet's Canvas Falcons(1929). By 1948 to chew someone out had become common during World War II meaning someone was given a verbal going over as in, "The Drill Sergeant chewed him out saying he would have his guts for garters because he was nothing more than a goldbricking lollygag." The colloquialism may have arose from Article 15, a form of non-judicial punishment from the Uniform Code of Military Justice. While it's not a full blown court martial an Article 15 can result in consequences ranging from an apologetic response to fines or confinement in the stockade.
MILREF: 2005 Archive of Military Definitions of the Day: