The speaker of "The Wild Swans at Coole" talks about twilight, October, autumn—all things associated with the end or latter part of things (the day and the year, respectively). He also flat out says that the swans' hearts haven't "grown old." (Unlike some speaker's hearts that we might mention.) It's pretty clear that he has old age on his mind, which is sort of weird because Yeats was only about 52 when he wrote the poem. (Come on! That's not exactly a time to move into the old folks home, W.B.) In reality, the speaker is really more interested in feeling old, in the sense that he has seen lots of things change. He's lost his innocence in a way (history/biography note: World War I will do that to you). For him, old age is really more of a state of mind.
Although the speaker has his reservations about getting older, he also suggests that it has its benefits. He is less carefree ("lighter") than he used to be, but he seems to be wiser. It's a mixed bag, that aging thing.
The sophistication of this reflection tells us that the speaker is not really as old or as weary as he'd like us to believe (the fibber). His mind and perception are still energetic and youthful.