The poet is probably on a mundane, routine errand and only has simple promises to keep. It appears that he lives on a farm and has driven into some little village in a horse-drawn sleigh to shop for ordinary supplies. Since it is nearly Christmas, he probably picked up some gifts for his wife and children, and the gifts for his children would be things he had promised to get them. When he got married he made the usual promises to love and to cherish, and he still intends to keep those promises, or obligations, to his wife.
Presumably he has been to the village where he does his usual shopping and is now on his way back home. If he had been on his way towards the village, he would not have stopped to look at the woods. He would not have stopped because he had too many things on his mind. But now he has accomplished his errand and his mind is free.
His little horse would not have shaken his harness to ring the harness-bells if they were on their way out, but would have shaken the bells if they were on their way home, where the horse is looking forward to being unhitched and allowed to rest in a warm barn with plenty to eat. In saying that he has promises to keep, the poet is saying, in effect, that he has responsibilities to other people and cannot indulge himself by remaining longer to look at the beautiful woods filling up with snow.
Robert Frost made his poetry dramatic. He has been quoted as saying:
Everything written is as good as it is dramatic. It need not declare itself in form, but it is drama or nothing.
"Stopping by Woods on a Snowy Evening" can be read as just a pretty winter snowscape, like on a Christmas card. But it is dramatic because the reader senses that there is more going on in the poet's mind than he tells us. There is something suspicious about the poet stopping. Why does he say:
Whose woods these are I think I know.
His house is in the village though;
He will not see me stopping here
To watch his woods fill up with snow.
Why should he care if anybody sees him? Why is he especially concerned about being seen by the owner of the woods? What is he planning to do? Many people have read a "death wish" into this poem. They think the poet is contemplating walking out into those dark, deep woods and lying down to freeze to death. His last repeated lines, "And miles to go before I sleep," can be interpreted as suggesting that he is travel-weary and weary of life but has many miles yet to go and many commitments to fulfill before he can enjoy the luxury of death.