Dry sticks are the result of using broken glass on damp sticks. They are used in the Sea Slug quest to light a torch on the Fishing Platform. They can be retained after the quest, and they can be used at any time to light a torch.
They are the equivalent of a tinderbox, except these cannot be sold in the Grand Exchange or used to light logs. Rubbing together the dry sticks will not produce a lit torch. A tinderbox and dry sticks both use one slot in your inventory. If you already have a tinderbox, save space in your inventory and drop the dry sticks AFTER you have completed Sea Slug.