> And I, as card holder, don't need any men-in-the-middle when I pay at brick-and-mortar shop: such shop has account in some other bank and has payment terminal from this bank, not from PayPal. Why does online shop need Stripe, PayPal or other men-in-the-middle?
Most likely the shop DOES have an intermediary like Stripe and PayPal, just not Stripe and PayPal. There are other POS providers out there. They might also be dealing directly with a bank and using a 3rd party for the equipment and integration. But all that is different from the web. POS/brick and mortar is different from online. It's generally more secure and less prone to fraud.
> Why should I use them and not my bank directly? Why does this business exist?
You can. You can literally go to a bank and ask about opening up a merchant account. However, the overhead is much higher. Basically, you'll be opening up your financial history and have a lot of rule sand regulations you'll have to follow. And you'll have to do the integration work if it's not already done. And don't expect Stripe/PayPal levels of easy integrations.
And, on top of all that, you have PCI compliance you have to manage, and that's no small thing. On top of that, customers will be looking for PayPal/Stripe/Apple Pay/etc. Why? Because they are customer friendly and more convenient.
But you can 100% go the route of getting your own merchant account, going through all the paperwork, putting aside sums of money, integrating, taking on all the risk yourself, by all means.
Or you can slap PayPal/Stripe up and be done.