The Mailbox stores event rewards, energy sent by friends, boxes from defeating villains, and various other prizes.
Most of the contents (except Tsum boxes) can be extracted using the "claim all" button.
The Mailbox can store up to 99 received from friends.
When the amount of energy reaches its maximum, the oldest ones will be removed. Energy sent from friends longer than 30 days ago will also be automatically deleted.
It also seems that if the Mailbox is overloaded with other kinds of prizes, the amount of energy stored will also decrease.
Tsum boxes Edit
Most of the contents of the Mailbox (like energy or event prizes) can be extracted using the "claim all" button However, this is not true for Tsum boxes (e.g. the ones you get for defeating a Villain in Battles).
Since version 2.7.0 the process of opening boxes has been sped up significantly: you can tap to skip most of the animation. However, the whole process still has to be done manually, which becomes a strain for stronger players who farm a lot of villains. As such, it is no surprise that many players resort to external software or OS features to automate this process. There are ways to do this on iPhones and rooted Android devices.
On iPhones the players take advantage of iOS accessibility features that allow repeating a certain sequence of taps. The video below explains as to how to do this.
Video courtesy of @Link from Discord.
There is a number of free (as in "free beer") apps for Android that allow automating a sequence of taps. A popular choice is RepetiTouch, a very simple yet powerful program in which you can record a sequence of taps opening one box and then "play" it any number of times you want.
Please note, however, that this kind of app is a low-level hack, so most if not all those programs will only work if your Android device is rooted!