The speaker compares their own body to a desert that suddenly gets rain, and the strange thing is that the rain doesn't feel like relief. Instead of drinking it in, the body seems to wake up a dead river inside it. That river rises, breaks out of its full bed, and leaves.
The title, "Desertion," helps us read this. Something is abandoning something else. The river runs off "with horses," an image of speed and wildness, in search of "a better bed to sleep in." It's as if the speaker's own life force, or desire, or love, has packed up and gone elsewhere, leaving the body behind as an empty landscape.
The mood is odd and mixed. There's longing and a quiet sadness, but also a kind of wild energy, since the river is more alive than the desert ever was. The poem seems to suggest that the speaker feels both left behind and a little stirred by what has escaped.