Very good question!!
I will tell you the explanation that one day my father told me but I do not know if it is true or not ..
Years ago, the saying was used for medical reasons, the phrase made reference to that if you worried a lot about the problems and you continued investigating more in subjects that were not necessarily important, you could get sick.
But really personally I think that currently the saying refers to human beings are very curious, we always want to know more information, we always investigate, we always want to know a gossip, we always want to know everything and sometimes that information hurts us more without I can avoid it.
In the phrase they use the cat because that animal is usually always careful and observant, unlike human beings who often act on instinct.
As much as it costs us to accept it, we should not always know everything in its entirety, sometimes just knowing the general things we can live with tranquility so relax and stop "looking for the 5 leg to the table" as people say in my country
Regards!