The speaker in this poem gives no identifying details about himself, except the fact that he is on a journey with his horse and that he has made "promises" to others and still has "miles to go" before he can stop moving. Evidently, he has made some commitments which cannot be broken. Despite this, however, he has stopped for what he accepts is rather a fanciful reason—simply to watch the woods "fill up with snow."
It seems like the man does not do this often; his horse, he thinks, must think it very odd for them to be stopping without any apparent purpose. The horse thinks the behavior of his master is "queer." It is also cold and snowy, so the horse is probably impatient to continue on its way, because it is used to continual movement when on the trail with its master. The speaker, however, is captivated by how "lovely" the woods are and seems to wish he could linger longer.