Starting at stage 4, you will start to see treasure chests in levels. These are special mini-games where you will play a hangman-type of game. You will have a fairly large word to guess and you are given 9 chances to guess a letter.
If you guess a letter correctly, it will show up on the screen and will not cost you a turn. If you miss, you loose that guess and keep going until you either guess the word correctly or are out of turns and fail out of the level.
As you play the game you will earn the ability to purchase a special upgrade. For 500 gems you can upgrade treasure chests to contain better rewards. This wont help you unlock the achievement faster but will help you complete levels faster.
Lastly, After you have beaten a stage, you can see on the levels preview if the stage has a treasure chest in it or not. Just complete 10 of these chests, you can even complete the same one 10 times if you like but you will earn this normally as there are more than 10 chests in the game.
The stats page in the games menu shows you your progress towards this if you would like to keep track.