A bye (B-Y-E) is a free pass through the current round of voting. A player may choose to use a bye (if he or she has one) instead of writing an entry for that round only.
Contestants may be given a specific number to be used before a certain point in the game, for example Season 6 players had 2 byes until the game reached 100 contestants. In a previous season, a player had been awarded the limited power to grant byes to other players as one of the many twists in the game.
If a player has byes available and they miss a deadline, they are assumed to have declared the bye, even if they did not. Once their byes are used, if they miss the deadline again – they are automatically eliminated.
You may have come across the sport of baseball and the notion of “three strikes and you are out”. In this case it is “three times of not turning in an entry and you are out”.
It’s called a “bye out” because a contestant is “out of byes” (and chances).