The meaning of let some of my cats on the table can be inferred from the author and the context.
J.L. Austin was an English academic from a middle class background. His father was an architect, and later the secretary of St. Leonard’s School, an independent school founded by the University of Saint Andrew’s in Scotland. J.L. Austin served in British Intelligence in the Second World War. After the war, he took the position of White’s Professor of Moral Philosphy at Oxford.
The English middle class is notable for its table manners, although perhaps less so now than fifty years ago. A cat would not normally be allowed on the table, and even a person dining alone or with a close friend would see the action as a special privilege being granted to a special cat. “Well, Rum Tum, I wouldn’t normally do this, but up you come. There’s a good fellow...”
J.L. Austin’s use of language is harder to pin down. He wrote extensively on the topic (one of his books is called Sense and Sensibilia). Most of his works were aimed at highly educated readers whom we can reasonably assume were drawn mostly from the same class as himself. He may have written technically, but his colloqialisms would have been distinctly middle class.
How do do things with words was adapted from a philosophical lecture series given at Harvard in 1955. As a visiting academic from postwar Britain to the United States, Austin would have been on his best behavior.
Let some of my cats on the table should be interpreted as an aside given by a performer to his audience. The performer “steps out of his role” for a moment and speaks plainly about his feelings and intent, even though these may themselves be part of the script.
The words mean exactly what they say. The well-mannered academic is telling his well-mannered audience that he is about to say some things that “want” to be said (like the cats wanting to be on the table), but which are usually not said in this type of setting.
I would add to this that Austin was decorated by three nations for his intelligence work: Britain (an OBE), France (Croix de guerre) and the United States (Legion of Honor). He undoubtedly knew many things that could not be said publicly, and for more reasons than just good manners. So the idea of “letting some cats on the table” probably appealed to him in more than one sense.