It's not restricted to pitchers; every player has a meat hand and a glove hand, and my impression is that it's used most often of catchers.
The earliest use I've found is from 1911, in a short story by Charles Van Loan, “The Crab” in The Big League, 1911, where a first baseman praises his colleague:
“There ain’t a third baseman in the country who has anything on him when it comes to handling a bunt or a slow ball. He comes gallopin’ in on ’em, drops his glove to the ground, straightens up, and wham! I can shut my eyes and take a throw from him, because I know just where he’s going to put it—and it’ll never be on the ‘meat-hand’ side, either.”
That is, the throw always comes to the first-baseman's glove; he doesn’t have to reach across his body.
Hugh S. Fullerton defines the term in “The Baseball Primer”, The American Magazine, LXXIV (May-Nov 1912), p. 204:
Meat Hand—The throwing hand of a player, the term resulting from the fact that the throwing hand is bare while the other is protected by a glove or mitt.
But there is nothing in these to tell us whether the term was invented by a player or a writer.