Batman: What do you want?
The Penguin: Ah! The direct approach. I admire that in a man with a mask.
Young Female Volunteer: You're the coolest role-model a young person could have!
The Penguin: And you're the hottest young person a role-model could have.
The Penguin: Just relax. I'll take care of the squealing, wretched, pinhead puppets of Gotham!
The Penguin: I am not a human being, I am an animal!
The Penguin: You're just jealous because I'm a genuine freak and you have to wear a mask.
Henchman: Penguin... Killing sleeping children. Isn't it that a little ah...
[Penguin grabs an umbrella and shoots Henchman dead.]
The Penguin: No! It's a lot!
The Penguin: It's true I was their number one son... But they treated me like number two!
Catwoman: I am Catwoman. Hear me roar.
Answer: The film keeps it ambiguous. There are two ways to look at it. Scenario A- She literally died and is brought back to life by the alley-cats somehow, adapting some of their traits. Or... Scenario B- The trauma of falling and sustaining a head-injury, along with her paranoia after Schreck tried to kill her, has driven her mad, and she uses her connection with cats to build a new persona. (Which is supported by the fact that all the things she claims "kills" her with each of her "nine lives" wouldn't actually have killed her. Ex. Her one fall is broken by the kitty-litter truck. Max doesn't hit her in any vital organs when he shoots her. Etc.) Her destroying her apartment is her lashing out at all the things she used to hold dear- her vision of a normal life, etc. It's symbolic of her purging the past and embracing the future. (Plus, oftentimes when people throw fits, they'll smash stuff up).