This poem is a meditation on death, but it's not just about the act of dying – it's about how the speaker imagines that he will be remembered after his death. Will he be remembered as a poet? Or as a novelist? Or as a nature lover? Or will he be forgotten immediately and left to rot in his house until the neighbors came by to check on the smell? The relationship of the present to the past (and to the future) has a lot to do with memory.
The neighbors' reported speech in "Afterwards" is written in prose to emphasize the difference between their language and the poet's – between their observation of nature and the poet's heightened sensitivity to it.
The poet hopes to be remembered as a nature lover, rather than as a great poet, because the two are concomitant: poetry can only be written by those who are sensitive observers of nature.