They have a ton of flavor, first of all
It is mystifying to me that there was a time in America when people didn't eat copious amounts of garlic. Garlic is delicious. Garlic is an essential part of so many dishes. I often have too much garlic in the fridge because the prospect of not having any is so terrifying to me. I usually double the amount suggested in a recipe. I mean, it makes food taste good! But as long as I've been cooking, I've been discarding garlic's paper-like skin as soon as I'm done peeling or smashing the clove. Turns out, that's a mistake.
Garlic skin isn't trash—it's something you can cook with and get a lot of flavor out of. Keeping it not only cuts down on food waste, but can impart a whole new level of garlic into your dishes. If you're making a sauce that you're going to strain anyway, you can just smash the garlic clove and keep the skin on. If you're miniching or slicing the garlic, just throw the garlic skin in a ziptop bag and freeze it. The next time you make stock, throw it in with your vegetables and bones. You can even just include it in a bouquet garni in a soup, as long as you make sure to wrap everything up in a nice cheesecloth packet beforehands—there's nothing more annoying than trying to fish out loose ingredients from a pot of soup.
Garlic skin has health benefits, too. There's a lot of vitamin A and C in there, as well as anti-inflammatory phenylpropanoid antioxidants. Some cooks also use a mortar and pestle to break up garlic skin and add it to bread dough to impart a mild garlic flavor and get those good nutrients in your next loaf. Garlic skin is a big help too when you're roasting garlic—you can keep the head intact and just squeeze out the cloves as needed.
So go ahead and keep that garlic skin rather than putting it into the trash or the compost. After all, garlic is as good as ten mothers.