In whisper enabled games, a player can choose to whisper to another player (send private messages). However the game is programmed intentionally to have a 5% chance of being overheard by other players. A leak/leaking is equivalent to a whisper being overheard.
Players who overhear the whisper will see the sender of the whisper. Sometimes, they can also see the receiver.
Some people will try to force a leak to occur, for the purposes of spreading false information. With the addition of the leaking button, whispers may now be intentionally leaked on cue.