Being a non resident of the United States doesnt mean you cant open a checking account in the US. When you apply for a checking account you will be given an ATM card to use so you will have instant access to your money. When you pay for something using your ATM card you simply swipe the card in the machine and your money will be transferred giving the store your money for the purchase. Today all banking needs can be done by computers.
ATM cards have several advantages; access your account balance anytime, and anywhere, almost every place accepts debit/check cards. When you purchase something with your ATM card your purchase amount is immediately transferred into the stores account. This allows them to have instant to the money for your purchase as quickly as if you had just handed them the cash. If the amount of the purchase is not in your checking account your transaction will be denied. If you want to transfer funds from one account to the other it is easy to do from an ATM machine.
Other things you may do with your ATM card is buy dinner, make a purchase over the phone, or reserve a hotel room or other transaction that will require you to use a card to pay for...