Electricity costs are low because electricity is inherently cheaper to generate and distribute than gasoline. Big, centralized power plants enjoy economies of scale that allow them to be much more efficient and operate more cheaply. They can also use fuels that are impractical for mobile use (basically all of them). Cars don't use gasoline because it's cheap, cars use gasoline because it's dense. If you don't need density then you can use much cheaper stuff, like coal, natural gas, uranium, sunlight, water trapped in a big lake, etc.

If everybody switched to electric cars tomorrow, electricity use would go up about 30%. This is far from trivial, but it's not the sort of thing that would cause prices to spike.

Answering the grandparent, payment varies. At home, you obviously just pay it in your electric bill. When charging away from home, some chargers are free. Businesses often put them in to attract customers, and Tesla operates a vast network of chargers whose operating costs are covered by car sales. (These chargers only work on Teslas, and electricity costs are low enough that they can afford to provide lifetime charging for their customers.) Some chargers are paid, and that's typically done with an RFID card (either a regular credit card with RFID, or a card specifically for the charging network) or a smartphone app.