What was Big Willy Shakes going for?
Literally, the phrase "set my teeth nothing on edge" means to feel a tingling in your teeth. But we're guessing you knew that already. Figuratively, Hotspur is saying that nothing makes him more uncomfortable than bad poetry.
Since the men have just been talking about who can speak English (and who can only speak Welsh), Hotspur means that anyone who doesn't know English properly is worse off than a kitten or an ungreased wheel on an axle.
Basically, it grates him when people misuse the English language in poetry. He thinks the English and their special language are pretty high and mighty, and he wants everyone to know it. (We're not so on board with that one, by the way.)
It turns out Shakespeare wasn't the first to jot down this phrase. Wycliff's Bible (published way back in 1382) says "teeth on edge" too. Shakespeare took it and ran with it, adding a "nothing" in the middle to make sure we really got the point.