The question supposes a particular theory about literature: namely, that the role of the text is to describe a fictional world, and the task of the reader is to determine facts about that world. According to this theory, it is unsatisfactory for Frost to tell us that he thinks he knows whose woods these are, but not to tell us who he thinks that is: by teasing a fact but not revealing it, the poet is interfering with the reader’s proper task.
It should be clear that this theory does not work very well when applied to this poem. The reason that it does not work is that the poem uses implicit metaphor, a technique which relies upon ambiguity, and ambiguity relies upon lack of detail.
Metaphor is a rhetorical device in which one thing represents another. Wikipedia gives the example “All the world’s a stage” from Shakespeare’s As You Like It, where the representation (of the world by the stage) is explicitly stated. But metaphors also work implicitly, where the poet does not tell you exactly what is being represented, or how the comparison works. This leaves room for the metaphor to work in multiple ways, allowing the reader to bring in multiple associations.
To take a famous example, there is no pun, double syntax, or dubiety in
Bare ruined choirs, where late the sweet birds sang,†
but the comparison holds for many reasons; because ruined monastery choirs are places in which to sing, because they involve sitting in a row, because they are made of wood, are carved in knots and so forth, because they used to be surrounded by a sheltering building crystallized out of the likeness of a forest, and colored with stained glass and painting like flowers and leaves, because they are now abandoned by all but the gray walls colored like the skies of winter […] these reasons, and many more […], must all combine to give the line its beauty, and there is a sort of ambiguity in not knowing which of them to hold most clearly in mind. Clearly this is involved in all such richness and heightening of effect, and the machinations of ambiguity are among the very roots of poetry.
William Empson (1930). Seven Types of Ambiguity, p. 3. London, Chatto and Windus.
† From Shakespeare’s Sonnet 73: a metaphor for the branches of a tree in winter, which are themselves a metaphor for the love-lorn mood of the speaker.
In the case of ‘Stopping by Woods’ the woods are “lovely, dark and deep” and the speaker would explore them if he could, but he has “promises to keep”. This is a metaphor for any situation in which desire and duty are opposed: perhaps the woods represent a poetic career, but the speaker has to earn a living; perhaps the woods represent escape to a life of adventure, but the speaker has a family; perhaps the woods represent death, but the speaker has dependents to look after. The poem provides no steer towards resolving this ambiguity, so that the reader’s own ideas about the conflict between desire and duty, whatever they are, can mix with the image in the poem.
The ambiguity extends naturally to the owner of the woods, so that, for example, if you have a religious metaphor in mind (the woods represent death, or eternity, or monastic seclusion) then the owner of the woods might be God, and his house might be the village church. Or if you have a psychological metaphor (the woods represent the speaker’s life, or subconscious mind) then the owner of the woods might be the speaker himself, whose home is with the villagers to whom he has obligations. The richness of the poem comes from this cloud of implicit metaphors.
But the more facts we learn about a thing, the more it can represent only itself, and the less it is capable of metaphorically representing something else. The more that Frost tells us about the owner of the woods, the harder it would become to fit the owner into any metaphorical scheme of interpretation, and the more the woods would leave the realm of poetry and become a commercial forestry enterprise:
The woods were bought by Farmer Black
Whose cousin Bill’s a lumberjack.
He paid five-ten an acre for it—
’Twas thought a bargain, nine years back.
Bill says that when the woods are sawed
Nine hundred thousand feet of board
Will go to build three hundred homes
In Boston, Springfield, and Concord.