The Mysterious Monster Egg can be bought in the Shop. The egg can be bought for 
50, acquired from daily missions, Tower of Masters, attendance bonuses and Epic Boss drops from defeating the scout.
The egg comes with a 80%, 17% and 3% chance of hatching monsters of a specific grade. The higher the player's level is, the grade of the monsters inside will be better.
For more information on what monsters appear in what egg at each level, visit Egg Levels.