I want my girlfriend to get rid of her pet.
What don't you like about the pet?
It's really annoying, I'm tired of listening to the damn animal meowing
It is unfortunate you do not like the cat, but I am sure your girlfriend does. Telling her to get rid of it will make her upset.
I will never understand why my partner loves that animal so much.
You may never understand, but there are somethings people do in relationship for their partners. For you, it could be putting up with her cat and not forcing her to give it up.