Elements of the Pauli group are the n-Pauli matrices with $\pm 1$ or $\pm i$ on front of them. They all commute or anti-commute between them.
The Clifford group are element that preserve the n-Pauli group under conjugation.
Is there a link between the result of the Gottesman Knill theorem, and somehow the fact that if you only use gates in the Clifford group you can simplify the circuit using commutation and anti-commutation rules from the n-Pauli group.
I didn't go into the proof of this theorem, I just would like to see if there is a handwavy but intuitive way to understand it.