First: You can have kiosks (in central locations like metro stations) or you can have dedicated points like where a market collects the cash and they add the balance that way.
Second: The public transit company. we aren't splitting the atom here.
In London you can top up your Oyster card in many small shops. Maybe someone here can confirm what the financial arrangement is - my assumption is the shop doesn't charge the operator as they get the benefit of more customers spending on other items.
Are the credit card transactions free?
Cheaper than managing cash, which is the relevant comparison.