It probably goes something like this:

1. By the Hochster-Roberts Theorem it is Cohen-Macaulay
2. The canonical sheaf of the quotient is a line bundle, because the elements of $G$ have det=1.

1. By the Hochster-Roberts Theorem it is Cohen-Macaulay
2. The canonical sheaf of $\mathbb A^n$ is $G$-invariant (because the elements of $G$ have det=1) and hence it descends to the canonical sheaf of the quotient, which is then a line bundle.