How does '"Do not go gentle into that good night" show the inherent closeness and relation of humans and nature?

First, the poem makes use of a central metaphor comparing death to night. Conversely, light is associated with life so that the "dying of the light" that occurs when night sets in is another way of describing death. In this way, our life is connected to nature's light, and our death is connected to nature's night; the span of a human life is connected to the span of one day. The narrator says that "Old age should burn and rave at close of day" (line 2). This means that old people ought to fight when it comes to the end of their lives. In this way, then, the author depicts a strong relationship between our lives and nature.

The speaker says that "Because [wise men's] words had forked no lightning they / Do not go gentle into that good night" (5-6). He compares the power of their words to the power of lightning via another metaphor, suggesting that they believe they've had no impact and so they refuse to die quietly.

In the third stanza, the speaker says that good men cry and he compares, via another metaphor, youth to a green bay in the line: "how bright / Their frail deeds might have danced in a green bay" (7-8). These good men feel that they might have been able to do so much more good than they did in their youths and so they fight against death.  

In the fourth stanza, the speaker compares living fully to catching and singing "the sun in flight" (line 10). He says that even wild men do not die quietly because they feel they haven't lived as fully as they could have.

In the final stanza, the speaker uses a simile to compare the literal blind eyes of old people to "meteors" that blaze (line 13). When these "Grave men" realize that they could have been happier in life, they regret that they weren't more happy when they had the chance.

In every stanza, there is a comparison that likens some aspect of human experience, especially our experience of death, to some aspect of nature. And these occur in addition to the central metaphor comparing death to night. Taken together, these comparisons, which are key to understanding the work, link our life experience to nature.