Where and how do you get and use coils?

You can also get them in specific ruby purchases.

You use them in the Alchemists Den to pull potions in storage so they aren't destroyed. Anytime you pull a potion of lv5+ it's automatically destroyed and you lose it. Levels 1-4 aren't lost. A reason to modify which potion is active is maybe you don't want it applied to pirates anymore and rather it be on armada. But the level is on the higher side. Rather than losing it, where it would have been lost; you're able to use the coil to remove it to create an empty slot so you can use that potion in the future.