On the surface, "Stopping By Woods on a Snowy Evening" is a simple poem about stopping to look at a snowy wood and then having the narrator realize that he can't stay in the woods forever because he has other obligations that evening. However, like many of Frost's poems, it is possible to see several elements in this poem as symbolic.
The author concludes the poem by saying,
The woods are lovely, dark and deep,
But I have promises to keep,
And miles to go before I sleep,
And miles to go before I sleep. (13-16)
It's possible that this is just about an actual physical journey and the need to get home, but the reader could also interpret these lines as the narrator turning away from some sort of dark or secret desire. It's also possible that in the dark, still, cold wood the narrator has been contemplating something like suicide or depression, and that he chooses to turn away from both because he has many things to do before he dies or "sleeps."
If you are looking at this poem symbolically, then the interpretation of the horse's actions might also change for the reader. The narrator mentions the horse when he says,
My little horse must think it queer
To stop without a farmhouse near
Between the woods and frozen lake
The darkest evening of the year.
He gives his harness bells a shake
To ask if there is some mistake.
The only other sound’s the sweep
Of easy wind and downy flake. (5-12)
The horse is the only other being that is with the narrator in the woods, and the narrator imagines that the horse thinks his behavior is strange. He shakes his harness, suggesting that perhaps there is some mistake that's brought them here and prompting the narrator to return to the rest of the world instead of staying in the woods. Symbolically, the horse might represent the pressures of the outside world or the influence of society. The horse might also be a projection of the narrator's own conscience, questioning him about whether or not this is the correct move. Either way, whether looking at the poem literally or figuratively, the horse is the first reminder that breaks the narrator from his own deep contemplation of the woods and reminds him that he has to go back to the real world and its obligations.