Pokémon with Flame Body have a 30% chance cause an attacking Pokémon that makes contact to get burned. For multi-strike moves that make contact such as Fury Swipes, each strike has a chance to burn the foe.
Outside of battle, if a Pokémon with Flame Body is in the party, the number of egg cycles are halved. It shares this effect with Magma Armor. It does stack with itself or Magma Armor.