It is very likely that "curiosity killed the cat" is meant rather than "care killed the cat". That idiom means that being curious is sometimes dangerous.

No, the cat was not killed. The Frank family actually left a note for the neighbors that basically asked them to take care of the cat. To Otto's suprise, when he visited the house after liberation, the cat was well taken care of.

Because...................................................................................

You are curious about why the cat was killed, and that was what killed the cat!

Curiosity killed the cat.

The mouse was killed by the cat.