On the Treasure Chest Event screen, you will be shown the remaining time of the event, the rewards, and the highest-scoring player.
In the top-right, you will see the remaining time of the event. Once this expires, you will no longer be able to contribute to the event. You will then have 24 hours to collect any rewards that you earned.
The text just below the title will explain the objective (e.g. Make ASSAULT RIFLE KILLS to unlock rewards!) This objective can be achieved in any game mode. Tapping the "i" button to the right of this will provide further information on the current event.
In the centre, you will see the rewards. The numbers at the top indicate how much progress in the event you have to make to unlock the reward. All players will be able to claim the free rewards when they reach the relative milestone but only players that purchase the Treasure Chest Pass will be able to collect the rest.
In the bottom-left, you will be shown the currently highest-scoring player as well as your current score. You can tap on either score to be taken to the relative leaderboard screen. The player that finishes first in the event will be rewarded a medal.
Finally, you will see the Treasure Chest Pass purchase button in the bottom-right corner. You can tap this to make the purchase which will allow you to be able to collect all the unlocked rewards in the event (free rewards can be collected without the pass).