Someone explain this to me. How exactly does the notched pickaxe raise smithing ability?

Easy, equip it and you gain 5 points in Smithing, up to an overall cap of 100.

You have to have it equipped to get the boost.

So if your smithing is 100 already it's somewhat useless.

Skill raised above 100 still works. Most will continue to work until 400. When I said work, it means bonuses associated with higher skill is applied to greater effect.

Unfortunately, the Notched Pickaxe itself is faulty, because I do not see any raise in the damage value of weapons I upgrade, with or without it. I suspect the reason is because it is a weapon, instead of an armor.

That isn't how upgrading items works, there are levels that you can upgrade your items to. You need to be a certain smithing level to reach a certain item level.

http://www.uesp.net/wiki/Skyrim:Smithing - explained quite well there, scroll down to 'item quality'.