How do banks manage and record very large volumes of transactions and customer accounts?

As everyone has cited, its predominantly computers. Add to this the divide and conquer rule. Essentially each transaction (in most cases) is a two-party exchange, i.e. a transaction between the payer and payee. So on an individual level, not much is happening, a ledger gets debited, whilst another one get credited. This is what the core banking software is really really good at. It understand what rule sets to follow (which have been defined in the system on a global level by default and specific rulesets as defined by the bank and the regulator they are working with).

Now scale this system to handle 10,000s of transactions per second and you will see that large volume handling is not a problem. Most core banking software will also act as routers (in the crudest of sense), pushing a transaction on to a particular platform for it to be executed. For example a credit card charge is handled by the CTL software. A mortgage payment handled by the Loans and Mortgage software. etc. So the transaction loads are off-loaded to other system.

The database itself is optimized to handle 10,000s operations per second as well.

The part that perhaps takes the most processing power in some sense are the checks and balances and log generation. Such software are combing through logs and transactions in real-time and then generating further alerts and logs on their defined rule sets.

Whilst the concept is explained here, simply, it is a complex system, but one if tuned perfectly, works without flaws within the defined operating ranges.