I don't think Angry Birds is popular because of its gameplay. Physics puzzles are a dime a dozen. I think it is popular because there's a profound and inexorable sadness at the heart of the game.
Because, look at those pigs. They've no limbs, and probably suffer from developmental disabilities-- certainly, their facial features exhibit some kind of congenital defect. They took the birds' eggs, but I don't think they understood that they were doing anything wrong; they were just hungry.
The birds swingshot themselves to death attemping to murder these crippled, developmentally-disabled pigs as a form of ultimately pointless and morally ambiguous vengeance. The yellow bird is graphically battered when it collides; the black bird explodes; the white bird drops an egg-- the very reason for their titular emotional state-- as a weapon.
It's not a deep game by any means; it is a bizarre one, though, and one that's very much a product of its time.