How do all the different banks know the money they receive in transfers actually exist since everything is digital?
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The inter-bank transfers are performed using trusted platforms. They don’t just take random emails from anywhere saying “Bank A sent $eleventy-billion to your account C, let them have that money”.
Lots of digital (cryptographic) signatures around to make sure everything checks out too.
Because the transfer itself is money. It is a promise from another bank to supply funds to that bank.
All any money is is a promise of future value. It does not matter if it exists in a physical form or not. And these days the overwhelming majority of money does not exist in any physical form. Heck, even in the early days most of it did not.