The proof of Fodor's lemma given on Wikipedia is quite the same as in my book and very understandable as well.
There is just one thing being ambiguous in this proof.
Why can one remove $0$ without loss of generality, i.e. why this doesn't affect on the structure of stationary sets?
Because $C=\kappa\setminus\{0\}$ is a club, so $C\cap S$ is again stationary.