Okay, let's start with taking the poem stanza by stanza and trying to determine what each means:
"THE TREES are in their autumn beauty, The woodland paths are dry, Under the October twilight the water Mirrors a still sky; Upon the brimming water among the stones
Are nine and fifty swans."
It's a beautiful dusk one fall day. The leaves are changing color and the sky is still. The waters of a body of water below are reflecting the pretty sky and the water has 59 swans swimming around in it.
"The nineteenth Autumn has come upon me Since I first made my count; I saw, before I had well finished, All suddenly mount
And scatter wheeling in great broken rings
Upon their clamorous wings.
It's been 19 years since the narrator started counting autumn swans on this lake. Before he/she even finishes this time the swans all take off in a clamor.
I have looked upon those brilliant creatures, And now my heart is sore. All’s changed since I, hearing at twilight, The first time on this shore, The bell-beat of their wings above my head,
Trod with a lighter tread.
Life is a lot different now, for the narrator, when he/she first spotted the swans on the lake all those years before. Still, they are so beautiful it is heartbreaking.
Unwearied still, lover by lover, They paddle in the cold, Companionable streams or climb the air; Their hearts have not grown old; Passion or conquest, wander where they will,
Attend upon them still.
Unlike the narrator, the swans seem never to age. They are still pared together with mates, paddling in the water or flying overhead. They still know the fire of passion and the challenges of conquering the world (so to speak.)
But now they drift on the still water
Mysterious, beautiful; Among what rushes will they build, By what lake’s edge or pool
Delight men’s eyes, when I awake some day
To find they have flown away?
They are glorious animals, but it is kind of sad...soon they will fly away and some other person will get to enjoy their glory while I am left alone.
That's about it, then (at least for my interpretation, perhaps you will find something different in the poem.) The swans are beautiful and free and seemingly ageless, while the narrator is human and aging. The swans come to remind the narrator how glorious life can be but then they fly off, leaving him/her alone. Kind of sad, really.