Before computers they called into a central reservations system and they looked it up for the travel agent.

This passage can describe it better than I could

To really understand the origins and development of the system we now call TPF we must take a trip back in time to circa 1940. We will visit a main ticket office of American Airlines in Little Rock, Arkansas, a growing company with growing ambitions. Here the basic control of flight reservation was a large card index file around which eight or so clerks would sort through the cards for the flight being requested.

They each knew the number of seats for the type of aircraft being used and by counting tally marks on the flight card they could tell if any seats were left and give you your “yes” or “no” on a reservation over the telephone. If your reservation was being made through another office it might take 2 or 3 hours to reach the revolving card index via a teletypewriter network and clerical personnel.

Insome of the medium-sized offices it was necessary to use binoculars to view critical information posted on large availability status boards in the ticket agent’s area. The absence of a red tag indicated that at least one seat was available on that flight. If more than one seat was needed a phone call to the back room might give you the availability which was again kept on three-by- five index cards according to flight number. Quite a system! Air travel was growing and it was obvious that this type of manual system was not going to be able to handle the business!

from http://www.ambriana.com/IT201_website/TPF_history.pdf

It also goes on with a story about how the SABER system was born