Stamps are obtained by completing milestones within the games. To fill the first ticket up requires 6 stamps. Simply play any non Xbox 360 title for the first time to unlock quickly (quit out, the achievement unlock is when you leave.)
An easier method only works if you've played any Xbox 360 game in the collection beforehand. For those, you need to have obtained at least 450 gamerscore from a single game (90 if the game is Banjo Kazooie, Banjo Tooie, Perfect Dark, or Jetpack Refueled). If you have, simply go to that game's menu screen to collect your stamps.