The bank acts as an intermediary, but not the way economists tell you. They say that the bank is an intermediary of loanable funds between debtor and creditor. That is, money exists outside the system and the bank is just efficiently distributing it, kind of like eBay.
Except the bank is an intermediary of a completely different kind between creditor and lender. What the bank does is aggregate illiquid credits and debts to create liquid credit and debt.
You have a coupon that says you are owed X products by person A. Person B has a coupon that says he is owed Y by person C. The bank takes these coupons and transforms these illiquid promises into a liquid promise that lets you buy both X and Y products from person A and C. Think of it as a many to many relationship. The bank essentially acts as a blender that takes many things of non uniform quality and it produces a product of uniform quality.
When you go to the bank and bring a stack of coupon that says "I will work for one hour for you" and the bank puts a stamp on the coupons that says "Bank B vouches for this coupon". Except this is inconvenient. What the bank does instead is print its own coupons that everyone recognizes and it deposits your coupons in its bank vaults. The bank writes down that you owe it all the coupons representing your labor time that you deposited as debt. Except this again is inconvenient, we can do away with the individual coupons entirely. Since paying your own coupon debt requires you to withdraw the coupons using bank coupons, we can just decide that you owe the debt in bank coupons instead.