How to hide fasteners on a horizontal wood fence?
Take a look at the category known as 'hidden deck fasteners'. It's typically a biscuit slot with a metal or plastic fastener that keys into the slot and then attaches to a post. (Or in the case of a deck, the joist.)

Top tip if you want to save time and wear on the biscuit joiner: have your lumber supplier groove both sides of your planks full length.