Okay, imagine if your mom asked you to clean your room. If you say, 'Okay, I will do it!' without asking why or saying you don't want to, you are doing it unquestioningly. It means you are doing something without asking any questions or thinking about it too much.
It’s like when you trust someone completely. So, when someone does something without doubting or asking 'Why do I have to do this?', they are doing it unquestioningly.