i am very confused about where the word "nothing" comes from. Did I miss something from the notes and tips?
You cannot translate every word. At least not always. The most direct equivalent of "She had nothing to drink." is "Neměla nic k pití".
"Neměla co pít." is a different way to say that. Literally translated it would be something like "She did not have what to drink.".
Czech learner here, I struggled with this "co pít" expression until it was explained to me thus: (affirmative): Měl co pít (he had what to drink). (negative): Neměl co pít (he didn't have what to drink).
("co pít" doesn't change! ).
I hope this helps.