Salesforce showing like mentioned in picture.
I assume that is a custom object of yours? In the section above your highlighted area, it says that the OWD's are "Controlled by Parent". That is why you cannot set sharing rules: whoever can see the parent record will be able to see the children adoption request records (provided they have access to the adoption request object).
What is it exactly that you want to do? What does the adoption request object represent? What is it a child of?