Note that these are just assumption based on observation.
To add to the other answers:
Good design can be born in many ways. Either by creation, but also by trial and error (or a mix of both). With airports people just realized that this setting works.
Note that this setting (2 floors) is typical in large airports. A good reason for having 2 floors is because it saves horizontal space and that is precious for airplane parking. Many small airports only have one floor because they have little traffic and enough space to keep everything at the same level (stairs, elevators, etc all occupy space and cost money).
Once you have the need for 2 floors you start thinking what to do with them. Since we have 2 types of airport users (arriving and departing) it's natural to keep each group together, at the same level. You could split the building differently, but using one floor for each group is, in principle, more rational.
Also we can easily assume, as others suggested, that airports worry more with departing passengers than arriving ones. Not that arrivals are not important but they stay less time and require less space. The comfort on departures is probably an important factor in an airport and it's where the airport can make extra money.
With this setting you can occupy part of the space in ground floor with technical areas, car parking, luggage handling and also arrivals. It;s also easier for employes working on the field to enter leave the building. And you might want to have those technical areas not only at ground level but also underground. This way you keep them adjacent.Having departures in the first floor allows not only for larger areas but also for natural light, either by using large windows or sky lights. Natural light and the sense of depth through the windows is a factor of comfort.
Probably as a consequence airport designers also realized that having the departures above puts passengers approximately at the airplane level and therefore jet bridges were born. I think they were born as a consequence but it's now a reason to support keeping the same design. It's also more comfortable to the departing passengers to just walk in to the airplane.
Currently in most airports I know departing passengers cross with arriving ones in the commercial area. But I've seen a few older airports where this didn't happen. Arriving passengers would actually be dropped on the ground floor. This kind of setting makes it easier to route persons (to add signs, etc). Also, you require less space since people always follow the same route and arrivals don't cross with departures. I now see many airports that drop passengers exactly in the same floor as departures. Although airports still try to separate them early, by routing arrival passengers down this is probably a commercial decision since it might induce consumption in the shops. After all many people still have to wait a couple of minutes for their luggage. Better have them around the shops than looking at a luggage belt.