How did curiosity kill the cat?

Curiosity killed the cat.  What an odd phrase. Yet a proverb.

Fear caused people to believe that being curious, being eager to know or learn something, was the cause of a living, breathing, harmless creature's death.

How could that possibly make sense?

Instead of encouraging originality, the intension is to scare children into keeping unique ideas to themselves?

Imagine all of the brilliance we have missed out on. These young minds have questions, but they don't ask because they think that if they do, a cat will die?

I understand the "mind your own business" translation to teach us not to gossip and worry about what other people are doing, but the communication seems more metaphorical of a larger, underlying problem.

Then I Googled "how did curiosity kill the cat?"