Wallace Stevens describes how an observer might respond to nature, something utterly different from the human experience, without pushing their own interpretation upon it.
For instance, the punchline in the third stanza, continuing off the first, is that one must have "a mind of winter" to observe what is literally in nature and "not to think / of any misery in the sound of the wind." Hearing misery in the wind would invoke a pathetic fallacy (Wikipedia), attributing the observer's own emotion to nature. It would violate the listener's desire (marked by the line break) not to think.
The trouble for the speaker is that humans cannot avoid metaphor (let alone thinking) entirely, so it's exceptionally difficult to describe nature without a human touch. So the last stanza uses wordplay with nothing to at least approach an outside-human description of nature:
For the listener, who listens in the snow,
And, nothing himself, beholds
Nothing that is not there and the nothing that is.
Nothing appears three times in three meanings:
Nothing himself - the listener after having negated his own ego, only listens and beholds, and only takes in.
Nothing that is not there - the listener takes in nothing that is not there; that is to say, he may only take in what is there. So he invokes no pathetic fallacy - no misery in the wind.
The nothing that is - the listener finds something else in nature. This is metaphorical, but unlike misery, it does not describe a person's feeling. Rather, the listener finds the absence of human mindfulness.
On this point, Anthony Channell Hilfer in " 'The Nothing that Is': Representations of Nature in American Writing" (Texas Studies in Literature and Language
Texas Studies in Literature and Language, Vol. 54, No. 2, Summer 2012) describes the poem as an attempt to negate the human presence when observing nature:
So the joke of 'The Snow Man' is that, finally, one cannot get around the pathetic fallacy. Yet the poem does go in the direction of doing so, chilling out human presence to something at least approaching zero-degree reality, an empathy with Nature exhibited more as a profoundly felt absence than as a presence. It does this by explicitly repudiating pathos as a proper response; pathos would involve the error of beholding "something that is not there." But the nothing that is has an evident sublimity, especially in its reduction of the human onlooker. (p. 231)