When you are in the game you earn the full branch profits (in coins per second) of your current bank plus the branch profits of all of your previous banks since you last prestiged. When you are away from the game, you earn a percentage of your current bank's profits for up to the length of time listed in the bank. Upgrading the bank lengthens the time you will continue to earn this percentage while away. Each time you leave the app the away timer is reset. i.e., if your away savings is upgraded to 2 hours you will get 10% earnings for the first 2 hours you are away from the game. When you launch the app you will be award these earnings. Each time you leave the app again you can get up to an additional 2 hours of away earnings.
Note, when you move onto the moon cities the earth bank regional profits do not carry over.