To bust out is to stop playing in a tournament due to losing all one's chips.
Note that bust out generally does not mean solely to lose all one's chips; it also means the player is no longer going to play in the tournament. In a tournament that allows rebuys, for example, it means either that the player is no longer allowed to rebuy or chooses not to do so. Busting out does not imply anything about whether the player has won a prize or not: in a standard tournament, all players except the eventual winner will bust out.
Busting out can also be used as a transitive verb, sometimes without the "out" part: players speak of "busting" another player, which means winning a pot in which the player who has gotten busted had placed the last of their chips. If the last of a player's chips go to another player, the other player is said to have "busted out" the first player, or sometimes just "busted" them.
Occasionally, busting out can refer to losing all one's chips in a cash game, but that is much less common than the use of the phrase to describe tournament play. Without any other context, busting is assumed to be happening in a tournament. Losing all one's chips in a cash game is more frequently referred to as being felted.