(third-person singular simple present chews the fat, present participle chewing the fat, simple past and past participle chewed the fat)
The origin is debated just as with many idioms. One of the possible origins is from the the fact that after a meal people would go into their sitting room with their guests and chit chat while chewing on the fat left over from a meal.
Another possibility comes from sailors who often had salted beef and pork on long voyages and the fat would harden. Sailors would eat and complain about the hard life of the sea while "chewing the fat"