As @MarkGrant pointed out the free condition is indeed the one that guarantees local triviality. The key point is to notice that every orbit of a point in $X_{n} - X_{n-1}$ has a unique representative in $D_{n} - X_{n-1}$. This way the trivializing cover would be the open sets of the form $p(X_{n} - X_{n-1})$. In my opinion the definition of a $G$-resolution becomes clearer when thinking of Milgram's classifying spaces. It also would be nice to know where one could find the mimeographed notes

@AndrejBauer I'll check it, thanks.

@abx yes, thank you. But I would like to know of a specific example.