What does it mean and how can I fix it?
zsh compinit: insecure directories, run compaudit for list.
Ignore insecure directories and continue [y] or abort compini
This answer is mostly a reference for myself to use in the future, as most answers do not provide a full-fledged solution. Here it is:
First run:
compinit
For every single path that is printed run the following the commands:
sudo chown $(whoami) PATH_HERE
sudo chmod -R 755 PATH_HERE
Simple example, let's say one of the paths that gets printed after running compinit is "/usr/local/share/zsh". Then:
sudo chown $(whoami) /usr/local/share/zsh
sudo chmod -R 755 /usr/local/share/zsh