I'm guessing you mean before computers, before buying stuff online, before everyone had a device in their pocket that can talk to computers and satellites, etc? (If not, what do you mean by "before everything was digitized"?)
Companies who could afford it had little credit-card machines that were hooked to their own phone line. This was just used to authorize a purchase, to see if you had enough credit for what you were purchasing. Then they'd take your card, run it manually through a device that would take the 'impression' of your card - this is why your name and card number were raised letters, which some cards don't do anymore - and it would record your information onto duplicate or triplicate slips: they kept one, one got sent to the credit-card company to make the actual transaction take place, and you got one. After credit-card theft became a big thing, they'd usually ask you if you wanted the 'carbons', the black sheets of carbon paper in between the pages that transferred the card info and your signature onto the 2nd and 3rd slips. Otherwise they'd just throw them away.
Stores that didn't have the little device with its own phone line would call the credit-card company on a normal phone. I have a vague memory that when I was very young - when there were more rotary phones than pushbutton phones, and dinosaurs roamed the earth - they would sometimes be talking to a person, but usually they were just pushing buttons: your card number, the amount (again, this was just to approve the transaction, no money actually moved until the companies sent the bundles of slips to the credit-card company), and some special code that I assume was, basically, the store's PIN number. If everything checked out - i.e. if the amount of credit you had, minus the amount of other approvals they'd given that month (which could tie up that money for a while if you ended up not buying the thing, because the credit-card company had no way of knowing until they got the slips), minus the amount of the current transaction, all added up to a number bigger than zero - they'd be told, then they would pull out the little manual device to take the impression of your card in triplicate.
Later - sometime in the 80s, I think - cash registers had slots in them for running the card through; there was still a lot of button pushing, which I assume was the current transaction amount and the store's identifying number/PIN.
It took time to do all of that crap... it used to be that the 'express lines' in grocery stores were for small, cash-only transactions so you could buy something quickly without standing behind a bunch of people paying with cards and checking IDs and signing receipts papers. Nowadays it's faster to use the card.
And now we have little jobbies you can plug into a smartphone that read cards and do the transaction over the internet in real-time, or you tap your phone onto something and it transfers the money; the hardware store takes Paypal and Bitcoin. A while back someone outside asked me if I could spare some change, and I stood there for a minute trying to remember the last time I paid cash for something... it's been a few years since I've used cash in the US. I have remind myself to grab a few bills for tips when I go out. (When I worked in restaurants cash tips were instant money, credit-card tips were maybe-someday money... so I always leave cash tips, even when paying the bill itself with a card.)