Good art either makes the strange common or the common, strange. It all goes back to ostranenie, the Russian neologism meaning "enstrangement" or "defamiliarization"--making the new seem old, the old seem new, as first detailed by Viktor Shklovksy. Some stories strive to do both, and I think the zombie story's in this category.
First, it takes something strange--a monster--and makes him common. By the end, hopefully, the reader feels for the zombie. He's become an everyman, could be a homeless person or a wanderer who's given up on appetite and decided to disconnect from the world and its suffering and pass on into whatever comes next.
Second, as a result of the monster becoming common, the type of story suddenly becomes strange. The typical monster story assumes fear of the monster; but when we begin to yearn for the monster to survive and even thrive, the story moves into less typical territory. In this way, the story takes a common cultural trope and enstranges it. This move, this sympathy for the devil, is a simple inversion. We're taught to fear monsters, but what happens when we get to know the monster as one of us? Frankenstein, Quasimodo, the Phantom of the Opera.
A somewhat contemporary literary example is John Gardner flipping the Beowulf story on its head and telling it from the monster Grendel's point of view. Same story, same concept is also done magnificently in Sturla Gunnarson's Icelandic film
Beowulf & Grendel from 2005, which develops a powerful sympathy for the monster. And the film
The Others from 2001 works along similar lines, though more subversively.
And as the cruel world closes in on the monster, as it inevitably does, and he becomes the hunted one, sympathy-for-the-devil moves the reader away from fear of the monster to fear for him. Speaking ideally, if the story functions on its highest level, the common or expected emotion of fear transforms into the rarer, more poignant emotion of loss. The zombie and reader are made to sense all that has been lost in his transformation--fly fishing, love for his dog, etc. And here, also ideally, the reader becomes linked to the zombie in a common sense of injustice.
*Key & Peele's White Zombie skit also enstranges the zombie trope by inverting the common direction of fear between zombies and people. (Notably, though, they don't make the zombies common; the zombies are still zombies in this sketch, strange and monstrous). When the White zombies show fear of Key and Peele's characters because they're Black, common tropological expectations are upset and, brilliantly, the very strangeness of racism is exposed, the ugly faces of the White zombies charged with it. In this case the enstrangement is made to carry the comment on racism.