He will not see me stopping here To watch his woods fill up with snow. (3-4)
Choice #1: our speaker decides to stop and watch the snow fall on trees. Why is this a significant choice? Well, it's getting late, it's cold, it's snowing, and he's still a long way from home. It may seem like a docile, serene scene, but we're worried he's going to get frostbite if he stays too long. Get it? Frost-bite? Hehe.
The little horse must think it queer To stop without a farmhouse near. (5-6)
Now we get a little more information about our speaker's choice to stop and watch the woods. Allegedly, he's chosen to stop at a place that is far, far away from any other humans. What does that say about him? We also would like to point out that our speaker is trying to figure out what his horse is thinking, which means that he must be kind of aware of how random it is to be stopping in the middle of nowhere.
He gives his harness bells a shake To ask if there is some mistake. (9-10)
You ring those bells, horse. We would too. Why do you think Frost uses the word "mistake" here? What mistake could the horse possibly be referring to?
The woods are lovely, dark and deep. But I have promises to keep, (13-14)
Choice #2: our speaker decides to continue on his journey home. It would seem that this is a particularly difficult decision for him, because the lovely and dark woods almost win out over the promises that must be kept.