Credits are what fund Minethings. They can be bought in the shop. Credits are used to buy and rent mines, and to get more inventory space.
The shop is where you can buy and rent mines, purchase inventory expansions, and purchase battery extensions for credits.
Mines: Mines can either be bought or rented in the shop. Bought mines will remain yours forever unless you sell them or get reset. You can also buy mines from other players for gold, although buying them for credits is much faster and easier. Rented mines cost 1/10th the cost of buying mines, but only last a week, and can't be sold for gold. The cost of buying a mine depends on the mine.
Inventory Expansion: The inventory expansion permanently increase your inventory limit by 25, with a maximum inventory limit of 350 (500 with a warehouse and avatar bonus). Your inventory limit starts at 75, and it can be very useful to increase this to be able to own more items. You can also buy warehouses (a gadget) for gold from other players, but these are temporary and only increase your inventory limit by 175. Each expansion costs 9 credits plus 6 credits for each one you had purchased beforehand.
Battery extension: This will temporarily increase your battery life by 10 days, which is helpful if your going to be away from a computer for awhile, and don't want your mining robots to stop mining. Extensions cost 9 credits each, and can stack on top of each other.