Proof of Work vs. Proof of Banking License
The Bitcoin system is a decentralized, distributed network of ledgers (nodes) that are continuously synchronized, validated and protected by a network of miners using proof of work.
Decentralized systems allow users to transfer value in a peer-to-peer fashion without the need for an intermediary such as a bank.
Decentralized systems are much more difficult to hack as you would be required to hack each node independently at the same time to be successful. Compared to a centralized system like a bank, a nefarious actor would only need to hack the primary centralized node to succeed.
Proof of work is a consensus mechanism to ensure the network that a party (the miner) has expended effort, energy, or resources to be eligible to validate transactions. Miners are rewarded with Bitcoin for their work.
Bitcoin miners fulfil the role of a bank, which is to validate transactions and prevent double-spending. Bitcoin miners do so by solving complex mathematical problems. By solving complex mathematical problems, miners have proven that they have expended effort, energy and resources, hence the name proof of work.
Proof of work is critical in maintaining the security and integrity of the network because, without the expenditure of effort, energy, or resources, anyone can attempt to maliciously manipulate transactions on the Bitcoin network and steal coins from users.
As an analogy, banks are heavily regulated and require banking licenses to operate, i.e. to validate transactions in our economic system. As a result, banks must expend effort, energy, and resources to acquire and keep their banking license. Another way to look at this is that the banks use the ‘proof of banking license’ or ‘proof of regulation’ consensus mechanism to prove that they are worthy of validating our transactions.
Banks are paid fees by the users for their services, and Bitcoin miners are rewarded with Bitcoin for their computational power.
In conclusion, you would trust a bank that their records are accurate and uncompromised because you know they have gone through all the effort to acquire and maintain their banking licenses. If the banks were to act in a nefarious manner, they would stand to lose a lot. Similarly to the Bitcoin network, you can trust that the records are accurate as a lot of work and energy are expended in securing the ledgers.
I would recommend that you read the full Bitcoin whitepaper in order to get a detailed understanding.