A gust of wind sets a hillside of trees moving, and the speaker stands in the middle of it. The opening image is playful: the trees "sway like drunks," lurching and loose, while their branches clack together like a noisy, wooden chatter. Nothing about the scene is calm, yet the feeling isn't anxious. The movement surrounds the speaker rather than threatening him.
The poem turns on its last lines. The speaker mentions, almost in passing, that his head cold is nearly gone, and that small detail changes everything. Being able to breathe clearly becomes a source of wonder: "How marvelous it is / just to breathe." The mood is light, grateful, and a little amused. Judging by the title, "Clearing" may refer both to the speaker's cleared head and to the open feeling of the moment, though that is an inference. What the poem seems to say is that ordinary relief can feel like joy when you notice it.