why does the cat kill rats?

The Nigerians know, for they have a folktale telling us why:

Ansa was King of Calabar for fifty years. He had a very faithful cat as a
housekeeper, and a rat was his house-boy. The king was an obstinate, headstrong
man, but was very fond of the cat, who had been in his store for many
years.
The rat, who was very poor, fell in love with one of the king's
servant girls, but was unable to give her any presents, as he had no
money.
At last he thought of the king's store, so in the nighttime, being
quite small, he had little difficulty, having made a hole in the roof, in
getting into the store. He then stole corn and native pears, and presented them
to his sweetheart.
At the end of the month, when the cat had to render her
account of the things in the store to the king, it was found that a lot of corn
and native pears were missing. The king was very angry at this, and asked the
cat for an explanation. But the cat could not account for the loss, until one of
her friends told her that the rat had been stealing the corn and giving it to
the girl.
When the cat told the king, he called the girl before him and had
her flogged. The rat he handed over to the cat to deal with, and dismissed them
both from his service. The cat was so angry at this that she killed and ate the
rat, and ever since that time whenever a cat sees a rat she kills and eats it.

Source for this tale here.