Note: Tracking the origins for the majority of popular idioms isn't easy, in fact, it's often impossible. Usually what you will find are plausible theories that try to explain a phrase's origin, but remember, they are just theories.
The quotes on this website come from older books, poems, newspapers, etc. They can be used as an indicator to determine how old some of these sayings are. Keep in mind, however, that if a saying is being used in an old newspaper, it's probably already well-known, so it's likely older than that.
knowyourphrase.com - Common Phrase Origins - Sayings and Their Meanings
This is a common expression that simply highlights a cat's curious side, which comes
as no surprise to anyone who owns a pet cat. Seriously, as an owner of several
cats, these little felines want to know about everything! However, sometimes their
curiousity can lead them straight into danger, and that danger might even involve
them risking or losing their life!
This idiom has been found in writings near the beginning of the 20th century, such as the 1909 short story Schools and Schools, by O. Henry. Included in the story is a line that reads:
"Curiosity can do more things than kill a cat..."
However, there is an older phrase that goes: "Care killed the cat." The 'care' is typically defined as 'worry' or 'sorrow'. This expression is rarely heard that much today, but it looks like there was a time when it was relatively popular. This older proverb goes back to at least the 16th century. It's used by a few playwrights, one of them being William Shakespeare; he wrote the idiom in one of his plays. For example, in the play Much Ado About Nothing from 1599, Shakespears wrote:
"What, courage man! what though care killed a cat, thou hast mettle enough in thee to kill care."