You get Influence by storing items in your supply lockers and completing machines. In return, you can use this to get things back out of the supply lockers, use the radio systems, and activate various base commands. Just take note that there is a negative feedback when getting an item back out of the supply lockers.
Your Influence will reset every day in real-time to regulate the amount that you can get in-game, and you are tasked to gain it back. There is another feature called Fame which you will receive by finishing missions and events. Fame is the bucket limit for Influence, and your Influence cannot go over the amount of your Fame. Both Fame and Influence are carried over between survivors.