For instance, "The grass is always greener on the other side of the fence" doesn't literally mean that your neighbor's stuff is always better than your own...just that we WANT it more than we want what we already have. Humans, as a species, have trouble being contented with our lot, no matter how good that lot may be.
But a few years ago, someone proved that for reasonably equivalent lawns, the grass really does look greener on the other side of the fence. When you look straight down, you can see between the blades of grass to the dirt or dead grass underneath, features that are hidden from view by even a slight angle. So, no matter which side of the fence you stand on, the grass really is greener on the other side.
Turning that back around on the original metaphor, though, only highlights why the metaphor works. The things we already have, the blades grass below our feet, have flaws that we have had plenty of time to notice. On the other hand, the things we don't have seem better, because their flaws are as-yet-unrevealed to us.
So, it's not merely a good metaphor because it points out our covetous nature, but because even less-covetous people may unfairly judge the familiar against the new, due to a greater knowledge of the flaws of the familiar.