The stanza in question, from Thomas Hardy's poem "Afterwards", comes from the first stanza of the poem.
The poem refers to the fact that things are changing in the world. The phrase "the Present has latched its postern behind" refers to the fact that the present is now the past. It, having already happened, can no longer be considered the present.
The line "May month flaps its glad green leaves like wings" also refers to the fact that May is now over and June has arrived.
The last line of the stanza refers to the fact that "he" (the speaker) is the type of man to notice that time was continuous. The man would notice that May is moving on and that the present does not stay around for long.
The speaker is reflecting on the fact of how others perceive him after his death. This is notated by the use of the past tense.
Basically, this stanza refers to the fact that the man was cognative that life simply moved on with the change of the world around him. Much like the delicate silk of the spider which typically goes unnoticed, time simply passes.