Why does 9 qubit Shor code have distance 3?
Distance 3 means that an error operator acting on 3 qubits maps a valid code-state to another valid code-state. For the Shor code, an example would be a simultaneous phase flip on the 1st, 4th and 7th qubits ($Z_1Z_4Z_7$).

You can easily check that $Z_1Z_4Z_7∉S$, but that it commute with $S$, hence is part of $N(S)$.