In the first two stanzas, the poem immediately launches into its use of pathetic fallacy. The rainy weather mirrors the bleak state of the couple's relationship. The speaker says that "The clouds have given their all," as if they are exhausted, just like people, from raining. This in turn sets up a clear link between the natural environment and the mood of the poem's two main characters: the speaker and his partner. (A quick note: the poem does not specify the speaker's gender, but to make it easier to read this guide uses male pronouns).
The fact that the "clouds had given their all" hints that the couple, too, have reached some kind of end-point. Two full days of rain have fallen; rain, of course, is often associated with misery and sadness. But now the rain has stopped, and the couple have an opportunity to take a walk. The alliteration of line 3, in which three out of the four words begin with /w/, anticipates the description of the earth as "waterlogged" in the first line of stanza 2:
in which we walked,
the waterlogged earth
This repetition of /w/ sounds gives the lines a sort of trudging, heavy quality, mimicking the feeling of having soggy mud underfoot as you walk.
The second stanza, beginning with "the waterlogged earth," intensifies the opening pathetic fallacy into personification, making the link between the couple and their environment even more pronounced. Now, the "waterlogged earth" is described as "gulping for breath at our feet." The consonance of the two /g/ sounds is similar to the sound of a gulp, and the enjambment in stanza 2 creates a further sense of breathlessness as the lines topple into each other.
Of course, mud doesn't really need to breathe in the way that humans do—but describing it as struggling for breath relates to the way that the couple, metaphorically speaking, are themselves struggling for breath. Whatever they've been arguing about has nearly broken them, and so their relationship is in this sense close to death. This is the cue for the swans to arrive.