The main role of the horse in the narrative of the poem is to emphasize, first, how unusual it is for the narrator to stop and watch a snowfall, second, to emphasize how long the narrator stays watching the snow fall in the woods, and third, to highlight the silence in the woods. In the second stanza, the narrator says his horse must think it "queer," in other words, odd, that they have stopped in the middle of nowhere--"without a farmhouse near," between the woods and a frozen lake. In the third stanza, the horse shakes his harness bells, a sign of impatience similar to a child saying "can we go now?" To the poet, the horse seems to be asking if there is some mistake that they've stopped so long. The jingle of his harness bells also emphasizes how still the scene is: the only other sounds are "easy wind," and "downy flake." In other words, it is hushed and quiet in the woods.
The poem thus turns a simple "prop," the horse, into a character almost in dialogue with the narrator, highlighting how unusual it is for this man to stop and enjoy the moment in a busy life "with promises to keep" and "miles to go before I sleep."