Most of these comments are far from ELI5 answers.

At it’s heart this term just means keeping accurate and complete financial records. Typically this term would be used by a “bookkeeper”, or accountant, to describe the process of producing financial statements for a business at the end of any reporting period, e.g. a month or a year, and checking to ensure they’re correct.

In essence, they want to make sure all of the transactions for a period are recorded. When people used paper checks to make payments they would call this “balancing” a checkbook, making sure that a ledger, or list, of all the checks they’ve written corresponds to the balance in their account, or “book”.

The bank would not keep a detailed record of your transactions, e.g. who you’re paying with each check. A bank statement would only include a list of checks by check number, or numerical identifier. People would manually keep a record of all the checks they’ve written (who they paid and how much) and check it against their bank statement. That process, the verification, was referred to as “balancing”.

Accountants keep many lists, lists of sales made, lists of expenses, lists of payments made, lists of payments received, etc. They “balance” (verify) many “books” (accounts).