Alliteration adds musicality to the poem and emphasizes certain words and phrases. For example, in the very first line, gentle /l/ alliteration draws attention to the phrase "little life." The phrase sounds delicate and quick, emphasizing the speaker's humble existence.
In the next two lines, listen to the thick /w/ alliteration as the speaker separates things that will "fade" from things that will "last":
I winnowed what would fade
From what would last till Heads like mine
Winnowing is the process by which farmers separate the wheat grain from the chaff using airflow (which disperses the lighter chaff). The alliteration here subtly evokes the whoosh of the wind that carries the metaphorical "chaff" of the speaker's life away. This windy sound appears throughout the poem, in fact, as with the /w/ consonance in line 6 ("blew away") and the alliteration in line 14 ("Whether it was the wind").
There's also insistent /w/ alliteration in line 7, calling to mind the chilly winter air:
I went one winter morning
This alliteration emphasizes a meaningful turning point in the poem: in the midst of a symbolic "winter" (i.e., a difficult season of their life), the speaker returns to the metaphorical "Barn" where they've stored their "Hay" only to find it isn't there. Alliteration imbues this line with intensity, alerting the reader that something important is about to happen.
There are other alliterative sounds in the poem as well. Take the bold /b/ alliteration of "business" and "begin" in lines 16-17, which adds some force and energy to the speaker's declaration of intent.