One of the reasons why Robert Frost's "Stopping by Woods on a Snowy Evening" is such a famous poem is that it is open to a myriad of interpretations. For instance, the dark, deep woods may symbolize death, in which case the poet stops for a moment on the journey of life to think about death, though at the end of the poem it is not yet time for him to die. He still has "miles to go" before he sleeps.
Here is another possible interpretation. The poet's journey still symbolizes life, but the woods are a symbol of something rather difficult to describe (after all, if something is easy to describe, why not write directly about it, rather than wasting time with symbols?). We might call it the beautiful, the numinous, the mysterious, the transcendent experience that stops you in your tracks and makes you stand and stare. This experience might, in fact, be caused by the sight of woods on a snowy evening, but it is the experience that matters, not the woods. It might equally be caused by a sunset, or a mountain, or feelings of love or compassion. It is the sort of experience which almost everyone has but which particularly appeals to mystics and poets.
The horse symbolizes the common-sense reaction of those who are not having this extraordinary experience, who "think it queer" to stand and gape at a snowy wood. Common sense is sometimes called "horse sense," and a sensible person would naturally shake his head and think there must be some mistake. Even the poet is finally compelled to give in to common sense (not to mention the intricate rented world symbolized by the man who owns the woods) and continues on the journey he must take in order to keep his promises.