Originally Posted by Falls City Beer
Sounds like "bore" in the old sense simply transformed metaphorically to the 1768 sense. Someone perhaps who grinds slowly and imperceptibly slow through conversations, like an auger.
A form of metonymy.
I agree. I'd imagine the first use was something like "You bore me," as in "you grind away at me." It later morphed to "I'm bored" and then to "a state of boredom."