Now, now. Before you reach for your "Team Edward" tee shirt, that's not what we mean here. In this poem, the speaker refers to a few different kinds of twilight. He refers more than once to the time of day (either at sunset or before dawn). This is a short poem, though, so we can't help but think that twilight is important to this poem's message. Our speaker is getting older and thus approaching the twilight of his time on earth. Twilight signifies the end of the day, and in the same way implies that the speaker is reaching the end of his time on Earth. Bummer.
Line 3: The speaker describes a scene in the October twilight. Right from the get-go, we're put into this time of day where the sun's light begins to fade.
Line 15: The speaker recalls hearing the sound of the swans' wings at twilight. Two twilights in 15 lines? We get the sense that he means twilight both as a time of day, but now also as a period of his life in decline.
Line 22: The speaker admires that the swans' hearts have not "grown old." We can assume, from this, that his own heart has. He seems to be in the "twilight" of his life.