first of all i want to say that i don't like grizabella very much. 
because of that my interpretation of her is probably very negative, 
nevermind. 

she was once a very beautiful cat and because everybody make her feel 
it she became posh. in my story she is the mother of jellylorum and 
asparagus (the younger one, not gus!) and also of tugger. she gave 
birth to him and then left the tribe because she wanted to show her 
beauty all the world. isn't it cruel to leave a kitten all alone? 
that is the reason why tugger hates her as he does in the video (if 
glances could kill!)
            
she wandered around, but suddenly she wasn't protected anymore by her 
family and many terrible things happened to her. nobody knows what, 
she would never tell. but she realized that leaving her family was 
the worst thing she ever does. so she returned, but of course the 
jellicles rejected her. she understand it but didn't want to give up. 
in the end they accepted her back after they realized that she has 
been punished enough and send her to the heavyside layer to start a 
new, better, life.

- Dragonfly