normalization - Normality of an affine semigroup

Sorry, I did not get the above answer. Can somebody please explain me.

Suppose, $q.(p_0,p_1, cdots ,p_{n-1}) in S$, where $ q in mathbb N setminus 0$
and $(p_0,p_1, cdots ,p_{n-1}) in G(S)$, why is it clear from the above answer that
$(p_0,p_1, cdots ,p_{n-1})$ is already in $S$. Some of the $p_i$'s can be negative
as well. It is not at all clear to me. Thanks in advance.