Worked out how to equip tools, it's pretty buggy though.
When you first equip an axe, the axe is teleported to what I assume is the origin of the space, and floats there as if carried by a player.
When you first equip a hammer, it follows you properly, until you use it on an anvil which will freeze it in in it's place at the end of your swing.
When a tool is dropped after using it the first time, the parts physically separate, but moving the handle will make the head float in the same direction. The head still highlights, but can not be interacted with. The tool can be equipped again.
If you then equip the axe for the second time, if follows the player around properly.
If you equip a hammer for a second time, it acts just as before.
When a tool is dropped for the second time, the parts separate again (if they were fixed), and the handle no longer highlights. You can no longer use the handle for anything, or equip the tool.
Aside from equipping tools, I'm not sure how you're supposed to turn wood into boards, hitting them only seems to make firewood.