Pigsy Droppings are fertilizer items that look like a batch of 3 brown round clumps with yellowish stripes.
They can only be obtained from certain tamed Pigsy variants, and can be used to speed up the growth of Crops (Wheat, Crisphead Lettuce, Turnips and Horned Melons), tree Saplings (Elderwood Saplings, Cragwood Saplings and Ashenwood Saplings), Mushroom Spores (Brown Mushroom Spores, Red Mushroom Spores and Glowing Mushroom Spores) and Queen Bees.
Pigsy Droppings can also be put on display.
How to obtain
For this, you'll have to tame either a Pigsy, Night Hoglet or a Night Pigsy with a Taming Collar. Craft a Taming Collar in your Crafting Menu - to be opened by typing "q" as the default key, then equip it and aim it at a Pigsy for 10 seconds or at a Night Pigsy or Night Hoglet for 15 seconds.
Pigsy Droppings will not "drop" from Pigsies or their brethren when you kill them.
All these Creatures will all fight back when being tamed and might sometimes push your player character away in the process, such interrupting the taming process. Night Hoglets are the easiest to tame as they do not inflict much damage and their pushing force is weak as well. Globs of Goo, Stun Bombs and building traps will help you with taming.
Then you'll have to feed your Pet, whereby you should try to feed it it's favorite type of Food that is displayed in the Pet Window (activate your Pet by clicking it with the right mouse button as the default keysetting).
Some Pigsy types are content with only a Red Mushroom and some Night Hoglets with only a Brown Mushroom, so you might want to tame several Creatures until you find a modest one that's "cheap" to feed.
You may feed all Pigsy types with different kinds of food than their favorite one, but in this case their pet-harvest might not contain any Pigsy Droppings.
Pets can only be harvested from after you have fed them, but otherwise they cannot starve (or die from anything actually, except if they are dismissed by their owners or players with the necessary permission rank that needs to either match or outrank the permission setting of the pet).
After feeding your Pet, wait until the Pet has digested and is ready to be harvested. If it is ready, the Pet will display a Power Cell in its thought bubble and in its Pet window.
Then you can pull the Pet with your gauntlet (you won't even need to have any Power Cell equipped) just as if you were digging a block - hold your left mouse button while looking at the Pet with your mouse cursor.
After harvesting, do not forget to wash your Pet. Equip a Washer (to be crafted too) and hold left click again while aiming at your Pet to hose it. Only cleaned Pets are able to get hungry again, and only fed Pets can be harvested from - which means that you won't be able to harvest from a Pet again without cleaning it.
As said, Pigsy Droppings cannot be looted from Pigsies, Night Pigsies or Night Hoglets when killing or dismissing them.
How to use
To be more precisely, you can use Pigsy Droppings on Wheat Seeds, Wheat Seedlings, Lettuce Seeds, Crisphead Lettuce Seedlings, Turnip Seeds, Turnip Seedlings, Horned Melon Seeds and Horned MelonSeedlings, on Elderwood Saplings, Cragwood Saplings and Ashenwood Saplings, on Brown Mushroom Spores, Red Mushroom Spores and Glowing Mushroom Spores, and on Queen Bees.
To use Pigsy Droppings for this purpose, you will first have to use right-click to place Seeds, Saplings, Mushroom Spores and/or Queen Bees from any quickslot on their respective fitting substrate and take care to choose a suitable environment so that they won't be fallow.
If Seeds, Saplings, Spores or Queen Bees are fallow, they cannot be fertilized, and they will not grow into Crops, trees, Mushrooms or Beeswax no matter what.
Please note that not every biome is suitable for farming these vegetal natural resources. They will usually stay fallow if you plant/place them into very cold environments, on very high altitudes, into very hot environments or into otherwise unsuitable areas, like into Swamplands or too close to several units of Bog Water.
Crops and Mushroom Spores both need either Water, Mineral Water or only a little Bog Water placed directly adjacent to the block that you plant them on (can also be diagonally) as a moistening, while light is not necessary for them to grow.
For tree Saplings, moisture is not necessary (different from what ingame tips might claim), but Saplings have to be planted/placed on the surface so that sunlight can reach them and they will have enough free space above so that they can grow into trees. Tree Saplings need either Dirt or green Grass as a substrate, only Elderwood Saplings will also grow on Snow Cap blocks optionally.
Queen Bees will grow into Beeswax if they're planted/placed on uncorrupted blocks of Wood or tree Leaves (except for Dark Wildwood Leaves) in suitable biomes with a little space around them. They need neither moisture nor light.
Then put (a stack of) Pigsy Droppings into any of your quickslots, select that slot by typing its number or scrolling there and click your right mouse button (as the default keysetting for "placing") while aiming your cursor at the Seeds or Seedlings (first stage of Crops growth), or on the tree Sapling, Mushroom Spores or Queen Bee that you want to fertilize one after another.
You cannot fertilize Sprouts (the third stage of growth directly after Seedlings), and you also cannot fertilize any crop more than one time - so once a Seed has been fertilized, the Seedling growing from it cannot be fertilized as well.
You cannot fertilize "soil" like Dirt blocks, Grass blocks, Mud blocks or tilled land (created with a Plow from the aforementioned natural blocks), neither can you fertilize Snow Caps (that Elderwood Saplings can be planted on) nor any block of uncorrupted Wood or Leaves that Queen Bees can grow into Beeswax on, and you cannot fertilize liquids either.
Mind that Seeds, Saplings, Spores and Queen Bees do not require fertilizer to grow into Crops, trees, Mushrooms and Beeswax.
Instead, fertilizing merely will shorten the timespan that these vegetal resources will take for growing.
Other than that, Pigsy Droppings cannot be placed into the game world directly, but they can be put on display on/in display containers like Placemats, Stone Wall Shelves, Hidden Temple Altars, Galactic Grav-Chambers, Frozen Containers, Holiday Decorative Trees, Snow Buried Containers, Holiday Mantles, Flower Pots, Wood Planters or the like. Pigsy Droppings might look a little different depending on the display container.
Effects of fertilizing
Fertilized Seedlings will grow into Sprouts ca. 10 minutes faster than without fertilizer.
Wheat grows fastest (ca. 50 minutes) without fertilizer, Crisphead Lettuce and Turnips will need approx. 60 minutes, Horned Melons would need 91 minutes to fully grow into Crops without using any fertilizer. Fertilizing Horned Melon Seeds right after planting/placing them is the most reasonable use for Pigsy Droppings.
You also can fertilize tree-Saplings to spurt their growth as long as they are not fallow - which will be the case if you planted/placed them on Dirt or green Grass (or Snow Caps in case of Elderwood Saplings) in a suitable surface biome.
Fertilized trees Saplings grow at about 15 minutes faster. It will usually take ca. 15-16 RL-hours until a tree will eventually "sprout" from a Sapling all of a sudden.
Crops, trees, Mushrooms and Beeswax all grow in real-time even when you're not in the game, so sleeping on a bed ingame will not make them grow faster.