There is a longer (and less often heard) version of this proverb
killed the cat
, but satisfaction
brought him back." I prefer this one, but it is a later add-on to the phrase.
Most sources I can find say it was originally "care kills a cat," and was first used in the 16th century. "Care" meant "worry" in this case -- too much anxiety is bad for you. The "curiosity" version is first recorded in 1909, probably as a result of the changes in the most common meanings of "care."