[Taniwha]

Originally they existed because nothing was computerised.

You went to your bank (and it was just that one physical bank) because that's where the definitive paper record of how much of the money in the bank was yours was kept. You also had a bank book, your copy of that record. Every time you went to a teller for a deposit or a withdrawal the teller would go to the bank's file storage, remove your record and update it, and your bank book to match the transaction, then return the record to the file storage.

If you moved town, or across town you applied to the bank to have your records moved.

Your checks (if you live in a country that still uses them) had your branch number on them, banks would reconcile checks using that info so they could update those all important file folders.

[eru]

Keep in mind that in large parts of the US, branches used to be banned. Every bank was only allowed a single office in one single city.

If you moved town, you had to find a new bank. Not just a new branch.

Obviously, that was a recipe for fragile banks, and thus frequent banking crises.