The trees are in their autumn beauty, The woodland paths are dry, (1-2)
The emphasis on "autumn" and "dry" is important. We think of autumn as the prelude to winter, a time of change (just think of leaves). "Dry" reminds us of a drought or desert, something that doesn't have life. These images suggest that nature is on the verge of death, as if its tranquility has more in common with death than life.
Under the October twilight the water Mirrors a still sky (3-4)
The word "twilight" is associated with death and endings (as in, "the twilight of my life"), not just hunky teen vampires. The poem is putting us in the mood for the contemplation of death that is to come.
Upon the brimming water among the stones Are nine-and-fifty swans. (5-6)
Fifty-nine is a totally random number. It seems deliberately uneven. It really does seem like one of the swans is missing, that it has disappeared or even died. The uneasiness we feel here—the sense of absence we feel—suggests that the speaker is reluctant to speak about death explicitly.