The poem sets two things side by side: the plain presence of a dog and the speaker's own anxiety. The title, "Honest to dog," plays on the phrase "honest to God," swapping in a dog as the thing worth swearing by. That small joke sets up the idea that a dog is a more trustworthy kind of truth.
In the first lines, the speaker stands "before the true hand / of a dog's being." Calling it a "hand" is odd and tender, as if the dog's whole nature reaches out, direct and unpretending. Then the poem turns. Next to that honesty, the speaker feels "the void in things" and "the stress that will eat me up." The dog seems to make the speaker's own emptiness and worry easier to see.
The mood is quiet and a little uneasy. There is real comfort in the dog's presence, but also a sense of being worn down. The poem is very short, and that bareness adds to the feeling: nothing is dressed up, much like the dog itself.