Hamster turds were initially only dropped by Server Hamsters, a titan-like enemy that roams around the game, but a Super Hammy also drops hamster turds as well. Super Hammy drops 1 turd per million damage, which caps at 5, and could further be multiplied by multikills.

Hamster turds are used to unlock the special colour #4, for which you need 5 hamster turds, and you'll need a 100 hamster turds to obtain the Hammius bobble. Check Bobbleheads for more info on it.

They also have a pretty neat icon, you can check how many hamster turds you have in the Other tab of your Inventory.