Is the sentence right or wrong?
Bella: "I'll look after your cat while you're away."
Bella said she would look after our cat while we were away.
We use "would" as the past of "will" in the reported speech.
Examples
Anne says she will come → Anne said she would come
I promise I will write → He promised he would write
They are sure they will be on time → They were sure they would be on time