I feel kinda stupid for asking this, but I can't figure it out! When you but art from another player, how do you pay them. This is embarressing, but I would really appreciate the help!

Don't be embarrassed! You're not the only one to ask this question. I didn't know how at first either.
There actually isn't a way to directly transfer gold to another user on Sylestia. You have to do it through an item trade. When the artist wants their payment, they send you an item via trade with the agreed upon price as the CoD. All you have to do is accept the trade and they receive the gold. Then you can send the item back to them for free. :)

Thanks!!