After logging in the Magic Window, press the "Assign Reader Sections" button, and two boxes appear below. Start typing the reader's name in the Reader box, and a drop-down list of readers appears (if it's a common name, a lot of names will appear in the list, so continue typing until the list gets short enough to choose the correct name easily). Copying and pasting the name may also work (with Ctrl+C, Ctrl+V), but it's trickier.
In the Sections box, you enter the numbers of the sections that are claimed. You can enter multiple numbers like 1, 5, 7, (for 3 sections) or even 1-3, 6-9, 12 (8 sections)
You press "Add" to assign the reader. You press "Remove" to orphan the sections by removing the reader.
The wiki page may have more detailed instructions:
https://wiki.librivox.org/index.php?title=BC%27s:_How_to_update_the_Magic_Window#Signing_up_readers
Please ask again if something is unclear.
If the reader's name doesn't appear in the drop-down list, most probably it's a new reader and hasn't been added yet. You can add them with the "Add New Reader" button. If they haven't given a catalogue name yet, you can use their forum name (to save time). If they give a catalogue name later (and/or personal url), an admin can make the change. You may suggest that the new reader does the 1-minute test before recording their section, but, in my opinion, it keeps things organized if they are nevertheless assigned in the Magic Window like everyone else.