You get this achievement after banishing 200 Limping Dragons. To banish a Limping Dragon you need a Dagger, which you can either find/win or buy from the store (2000 gold). Simply click on a Limping Dragon and then press the button to banish (or to buy a Dagger and banish if you don't have one).
Limping Dragons will not unlock until well into the game (level 21 or higher).