This page of the Weird West guide explains how to obtain the Skinning Toolbag - an item necessary for skinning animals and obtaining valuable crafting materials.
Weird West lets you craft items if you have the correct materials. Animal skins are one such material.
Before you can skin animals and collect their hides you need the Skinning Toolbag, purchasable from tailors starting from the first town. The tailors sell other useful tools and items as well. Depending on discounts you'll pay around 70-80$.