WHEN the Present has latched its postern behind my tremulous stay,
The speaker starts off on a cheerful note: these first lines basically say, "When I'm dead…." Of course, Hardy doesn't put it so bluntly, and some of his language is pretty ambiguous. Let's look more closely at the language – the "Present" is shutting the back door ("postern") on the speaker's life.
The fact that time ("Present") is "latching" the back door on the speaker's life suggests that he's afraid that he'll slip away and won't be remembered at all.
"Tremulous stay" is also difficult – it refers to his "stay" on earth, or his life. Why would he call his life "tremulous," or "trembling"? Is it because, as an old man, the poet's hands aren't as steady as they used to be? Maybe, but it could also be more universal than that. Perhaps the poet is suggesting that all life is uncertain and shaky.
"Stay" also suggests the idea of structural support ("stays" are the ropes that help support the mast and sails on big sailing ships, and the word also describes other kinds of support structures, like in old-fashioned lady's undergarments). "Stay" implies structure and steady support, and "tremulous" means shaky, so the juxtaposition of "tremulous" and "stay" creates an oxymoron, or a contradiction in terms.
And the May month flaps its glad green leaves like wings, Delicate-filmed as new-spun silk,
The speaker has imagined that he is dead, but instead of following that up with a bleak wintry landscape, he says that it's springtime – this comes as a bit of a surprise.
The words, "delicate" and "new-spun silk" could describe the new spring leaves on the trees (which are, after all, very delicate and tender when they first come out), but "delicate" and "new-spun-silk" could also be paired with the "wings" – so the "glad green leaves" are as "delicate" as butterfly wings.
"Silk," the fabric of choice for many in the 1990s, is produced from the cocoons of the silkworm – so the "new-spun silk," although referring most obviously to the delicate, silky new leaves in the month of May, also suggests cocoons and metamorphosis from caterpillar into moth or butterfly.
There's a lot of movement in this description ("flap[ping]", spinning silk, new growth) considering that the speaker of the poem is imagining the world after his death.
[…] will the neighbours say, "He was a man who used to notice such things"?
The speaker wonders whether, after he has died, his neighbors will remember that he used to enjoy the springtime.
This question is repeated at the end of each stanza, like a repeating chorus or refrain. The question of what his neighbors will think of him after he has died becomes almost an obsession for the speaker.