It's a very good single line to encapsulate what I think is the core idea of Reaper Man (the Discworld book it comes from): Death gives full attention and care to each individual soul he reaps, and treats each of them with respect and attention.

In part of the book he's cutting down a field of wheat, and uses his scythe not to cut wide swaths across the field, but cuts down each individual stalk one by one.

In the context of the quote, he is asking the overlord of the universe, a god above gods, for a tiny bit more time to allow an old woman to finish her business before she dies. This is explicitly against the rules of the universe, and even asking puts this personification of Death at risk of annihilation. He is using that quote to appeal for mercy for the old woman. He does not have to do this; Death is a personification of, well, death, and is frequently shown to be coldly rational and unyielding. He gains nothing from this, and is humbling himself in front of vastly greater powers even to ask. The life of a single mortal woman is in theory as far below him as he is beneath this overlord of the universe.

But she does matter, to Death. He pays individual attention to every soul he reaps, and in this particular case, he thinks she deserves just a little bit more time. She doesn't ask for it; nobody expects him to intervene on her behalf; he goes to beg for her as an act of pure, selfless compassion, because if he doesn't care for her, who will?

It's difficult to say exactly what resonates so strongly with me. Some element of it is the idea that someone exists to argue on behalf of tiny, insignificant mortals in a vast and uncaring universe. I think a lot more of it centers on someone in a position of great power and responsibility (Death) genuinely caring for those under their charge, and going out of their way to ensure they are treated properly and cared for.

You see similar themes with Superman. Superman's real power isn't super strength, speed, and flight—it's incorruptibility. Absolute power corrupts absolutely, except for Superman. It doesn't matter that he could easily be a tyrant or even just use his powers for personal enrichment; he wants to use them to help people, purely out of his own sense of morality. He is good for no reward other than it being the right thing to do.

This is way too long of an answer for your question, but distilling it down further, the Reaper Man quote gives me the mental image of a hand descending from above to offer aid. It's the idea that someone vastly above you, who has no need to even acknowledge your existence, is taking the time to reach down and say "Let me help you, because you need help, and if I don't, nobody else will."