While you cannot edit someone else's public quiz, you can create a duplicate of that quiz and edit that copy freely.
Just go to the quiz you want to edit, and click the 'Edit' button (see screenshot below).
The quiz will be duplicated automatically and ready to edit from the Quizizz editor after clicking the 'Edit' button from the quiz page.
You can now freely edit this quiz, and you will always find it again in your "My Quizizz" section.
Note: That this is your copy of the original quiz, and changes you make will only affect your copy, not the original quiz.