Hey! You have to have a weapon equipped, so after you pick up the stick, press I to go to inventory, and click the stick to equip it.
You can then left click to swing it. It uses a collider on the stick, so you have to be attacking and hitting them with the stick to do damage. You can find other weapons, and they should work too.
There's not a lot of great feedback to tell you when you hit an enemy or not :(