In Japan there are cards (and, indeed, your cell phone) that work exactly the same way. Debit cards don't really exist in Japan. Instead you buy a smart card (often the ones used for paying train or bus fares). You put some money on the card using a machine (you insert cash and it encodes the money on the card). Then you can flash the card onto a reader to pay for things (often train or bus fares, but in the past 10 years it has extended to vending machines, convenience stores, etc, etc).
Lightning is exactly the same way. You set up a contract with the payment processor. This requires the transaction to be accepted into a block. Then you make as many smaller transactions (which require no blockchain transactions) with the payment processor as you want. At any time, the payment processor can terminate the contract by cashing in the amount of money you have spent. This requires a transaction to be accepted in the block chain. After that the channel is closed and only the amount spent is transferred (the remaining amount is freed up). If it goes beyond the contract length without the payment processor cashing in, then the contract is cancelled and the total amount is freed up.
So basically it allows you to make many transactions through a payment processor, while only making 2 transactions on the block chain. Anyone can be a payment processor, but the protocol requires the payment processor to have capital equal to the transactions in process (they have to transfer funds to a third party before they get paid -- although they are guaranteed to get paid by the end of the contract).
It's actually a pretty slick protocol. Their website has a video with implementation details that explains exactly how it works: https://lightning.network/