I silence birds and stop Mills,
Since birds mate in the spring, they don't sing until winter ends. Mills shut down in winter because there is no grain to grind, since it was all harvested in the summer/fall. Furthermore, if the mill is hydro powered, the river may freeze over and not be able to power the mill.
Hide hills and make rivers still.
Heavy snowfall can make hills seem flat (if the area around the hill is covered with more snow), and rivers can freeze over, giving them the illusion of being still.
Meanwhile, I'll make beasts slumber,
Animals hibernate in wintertime: examples include bears, squirrels, and skunks.
I cannot stay, but I may linger.
Summer will always roll around (Even when Winter Is Coming), but winter will last a while. Added by jpmc26: the word linger could also be a reference to times when it seems to last longer than other winters (i.e. Groundhog Day).