Fallout Shelter helps kids practice and improve the following skills:

Organization

Arranging and coordinating materials in order to complete a task.

Fallout Shelter requires the player to organize multiple things:  they have to choose which rooms to place people in and which clothing, weapons, and pets to give each person. They must also keep track of who is happy and who isn't, which people need leveled up or to have skills improved, and whether or not they are producing more consumables (such as electricity) than what their vault is using. The success of their vault relies directly on how well they use these cognitive processes. Without staying well-organized, people will be unhappy, the player's daily bonuses will be lower, and they will run out of resources. When this happens, not only will they be unable to expand their vault, but their vault will fail.

Self-Awareness

Understanding our own actions, thoughts, and feelings.

Fallout Shelter also requires the player to utilize their self-awareness skills. First and foremost, they must be able to analyze their vaults (as a stand-in for themselves) and evaluate whether or not they are performing at an optimal level. Just like in real life, this means they must be able to balance several areas of existence (resources, crafting, exploring, clothing and weapons, and population size). If they don't remember to examine (and adjust as needed) all of these areas, they will not create a successful vault.

Secondly, they must also exercise their empathy skills by examining the happiness of their vault dwellers. The more unhappy dwellers they have, the lower the vault's daily grade will be, and the fewer bonuses the player will receive. Fewer bonuses will greatly slow down the growth of the vault and, eventually, even lead to systems and vault failure.

Mathematics

Fallout Shelter includes a large amount of numbers and statistics in order to help the player keep track of the success of their vault. This provides the perfect opportunity to work on mathematics skills. Things like comparing food production with food consumption, whether or not moving a dweller to a different room will increase production, and even the best way to spend caps (Fallout's money system), all require the use of math skills.

Reading

While it is less prevalent, Fallout Shelter does require the use of reading skills as well. The player will have greater success if they are able to read the daily report, the objectives, the tips, and the stats, as well as being able to understand the descriptions of each room type and which dwellers are best for that room. All of this requires reading and comprehension skills. Without them, the player will have to guess what they are doing right and wrong, which will not lead to a very successful vault.