The phrase curiosity killed the cat and its variants mean: making unnecessary inquiries or investigations may result in unhappiness or misfortune; some things are better left unquestioned or undiscovered.

The earlier form of the phrase was care killed a cat, in which care means disquiet. It is first recorded in Much adoe about Nothing (Quarto 1, 1600), by the English playwright and poet William Shakespeare (1564-1616)—Claudio and Don Pedro are addressing Benedick:

Don Pedro: As I am an honest man he lookes pale, art thou
sicke, or angry?
Claudio: What, courage man: what though care kild a catte,
thou hast mettle enough in thee to kill care.

In Euery man in his Humor, first published in London in 1601, the English poet and playwright Benjamin ‘Ben’ Jonson (1572-1637) makes Cob use the phrase in a medley of exclamations successively expressing disordered haste, defiance of sorrow, aversion to worry, which is liable to kill even a cat, [uptails all is of obscure meaning], and a curse on the hangman:

He owes me fortie shillings (my wife lent him
out of her purse, by sixpence a time) besides his lodging,
I would I had it I shall haue it he saith next Action
Helter skelter, hang sorrow, care will kill a cat, vptailes all,
and a poxe on the hangman.

In Dictionary of Phrase and Fable (Philadelphia, 1898), the English clergyman and schoolmaster Ebenezer Cobham Brewer (1810-97) recorded the form care killed the cat only; according to him, the underlying image is that disquiet would exhaust the nine lives allotted to a cat—those nine lives were already proverbial in the mid-16^th century [see footnote]:

Care killed the Cat. It is said that “a cat has nine lives,” yet care would wear them all out.

The earliest occurrences that I have found of the phrase curiosity killed the cat and variants are as follows, in chronological order:

1-: From Aunt Hetty’s Stratagem, an unsigned short story published in the Waterford Mirror and Tramore Visiter (Waterford, County Waterford, Ireland) of Wednesday 28^th October 1868:

Often when I came home from a solitary ramble, I would find Aunt Hetty and Carrie in close conversation; and what caused my surprise, they became suddenly silent whenever I appeared. I was very curious to know the cause; I couldn’t help but suspect that it in some way concerned me.
‘Carrie,’ I said one day as we were taking our usual walk, ‘do you know you have begun to make me jealous? It seems to me Aunt Hetty takes you a great deal into her confidence. She never thinks of confiding in me so!’
‘Am I,’ said Carrie, laughing merrily. ‘I am sorry; but you will know by-and-by what Aunty Willis and I have so much to talk of.’
‘Very well,’ I said, laughing. ‘They say curiosity killed a cat once, so I will take warning and let things remain as they are.’
‘That’s very prudent, Hester,’ said Carrie tapping me playfully on the shoulders.

2-: From What Money Can Buy, a short story by Miss Mary Lee, published in True Flag (Boston, Massachusetts, USA) of Saturday 10^th July 1875:

While she was pouring out her father’s second cup of coffee came a ring at the bell. “A note from Mrs. Fay to you, Miss Belle,” Maggie announced, bringing in the note gingerly between her thumb and fore-finger, as though she were a little afraid of it. “And the man said there was no answer. Awful fine he was, too, in a livery coat. What should she be a writing to you for, dear? Has she any little girls to take music?”
“There, Maggie,” began Colonel De Silver.
“Curiosity killed a cat,” Belle said, carelessly, tearing open the note.

3-: From Notes and Queries, published in the New York Dispatch (New York City, New York, USA) of Sunday 24^th April 1881:

Snuffy.—“Could you kindly inform me of the origin of the phrase ‘Curiosity killed a cat?’” We have never heard the expression, and are of the opinion that you misquote it. The phrase is “Care killed a cat.” You will find it in the first scene of the fifth act of “Much Ado About Nothing.” Claudio says to Benedick: “Courage, man! What though care killed a cat, thou hast mettle enough to kill care.” It is probable that it was in common use before Shakespeare’s time.

4-: From A Kitten’s Perplexity, published in The Savannah Morning News (Savannah, Georgia, USA) of Friday 21^st September 1883—reprinted from the Chicago Times (Chicago, Illinois, USA):
—In this story, a pet kitten, owned by some children, sees its own image reflected in a mirror and mistakes it for an actual cat; the kitten bangs its head against the mirror when it tries to make contact with its reflection, and finally withdraws to the solitude of the cellar:

During the remainder of the day it kept away from the living rooms of the house, and when taken its usual meal of bread and milk it left the dish untasted. The shock to the kitten has had altogether such bad effects that its little playmates manifest much concern lest its queerness be the precursor of its death, and additional testimony thus be given of the plausible proposition that curiosity killed a cat.

 

Note: In A Dialogue Conteinyng the Nomber in Effect of all the Prouerbes in the Englishe Tongue (London, 1546), the English playwright and epigrammatist John Heywood (circa 1496-circa 1578) wrote:

A woman hath nyne lyues lyke a cat.