So I won some reputation, and now I'm '13'. But whenever I log in again, it shows that my reputation is back to '1'. The higher rank only reappears when I open the notifications and click on the note that says that I won some reputation. What is going on?
The "1 rep" is for the SE sites you haven't earned any reputation. Your reputations is shown on each SE site separately.
Look at this picture, from your profile:
So if you open Philosophy.SE, you've earned "1 rep" there and that's what you must see next to your display name.
If you open English.SE, you must see "11 rep".
If you open Linguistics.SE, you must see "13 rep".