Why do varieties with torsion canonical sheaf have finite etale covers with trivial canonical sheaf

This is a special case of a more general fact. If we are given a line bundle $L$ on an algebraic variety $X$ and an isomorphism $L^{\otimes n} \cong O_X$, then there is uniquely attached to this data an $n$-fold étale cover $Y \to X$ such that the pullback of $L$ to $Y$ is trivial. Note that the pullback of the canonical sheaf under an étale map is the canonical sheaf.

One way of realizing this is to use that $X$ is isomorphic to its image in the total space of $L^{\otimes n}$ under the unit section obtained from the isomorphism above. Now every point on $X$ has $n$ distinct "roots" in the total space of $L$. I think all this is explained in Griffiths-Harris, for instance.