Put a sock in it means quieten down and is a politer, more jocular, alternative to shut up or stop that row.
Its first appearance in print was in The Athenaeum on 8th Auhust 1918. That august (s well as August) magazine defined it as “Leave off talking, singing or shouting” and the fact that it had to be defined suggests it was a relatively new phrase. Its most likely origin is in World War One slang and the thing that the sock was to be put in was the mouth – like the earlier expressions ‘put a cork’ in it and ‘put a bung in it’.
There is a popular story that the phrase originated with the idea of putting a sock in the old style phonograph which lacked a volume control but there is no evidence for this and it appeared after the gramophone had already been around for some time.
This cat is trying to get the phrase ‘put a cat in it’ into the books of English idioms but I doubt it will catch on!