The poem is called "The Wild Swans at Coole," so it's no surprise that swans pop up all over the place. They are a symbol of beauty, but also of energy and permanence. They don't seem to age the way the speaker does, and they aren't conscious of pain and weariness in the way that the speaker is. Although the swans impress the speaker greatly, they also remind him of the pains of getting old and the inevitability of death. So, it's a mixed bag with these guys.
Lines 5-6: The speaker says there are fifty-nine swans on the water. Ever tried to count animals? They don't hold still while you do. Either this speaker is studying the swans intently, or he's so familiar with them that he knows how many there are to begin with. After all, he's been coming to see them for nineteen autumns.
Lines 10-12: The speaker recounts seeing the swans "mount" into the air. They make a lot of noise (their wings are "clamorous," after all).
Line 13: The swans are described as "brilliant." This suggests that they are luminous, or shiny. They seem to glow with youth and energy.
Lines 17-18: The speaker mentions the "bell-beat" of the swans' wings. It seems that the energy and motion of the swans also create an awareness of time (which is what bells toll) in the speaker.
Lines 19-21: The swans "paddle" in the water and "climb" the air. They seem to have an energy and lack of weariness that the speaker really can't identify with.
Line 22: In a synecdoche, the speaker tells us that the swans' "hearts" have not grown old. Presumably, though, the speaker's has.
Lines 23-24: "Passion or conquest" will still "attend" the swans, no matter where they go. In other words, in the speaker's mind these swans will always possess energy and exuberance. Swans don't really possess "passion or conquest." Giving human qualities to a non-human thing—that's called personification.
Lines 25-26: The swans are drifting on the water. They are "mysterious" and "beautiful," but it seems that they have, temporarily anyway, lost the great energy that the speaker admired in previous stanzas. We wonder if this isn't the speaker's effect on them.
Lines 27-30: The speaker wonders where the swans will build new homes and where they will "delight" other people—not him—with their beauty and spirit.