Why do we call it an apron?
Airport apron refers to the area of an airport where aircraft are parked, unloaded or loaded, refueled, or boarded and looks like this:

Image from airplane-pictures.net
This probably originated from the word defining theatrical apron:

1903 A. B. WALKLEY Dramatic Criticism 108 The ‘apron’ is the technical name for the stage-area in front of the curtain. In the Elizabethan theatre it jutted right out among the public, who surrounded it on three sides. This ‘apron’ slowly shrank..till at last in our day it has altogether disappeared.

As the airport apron looks similar to this, it probably go its name. The following image shows an apron stage (The Visy Theatre in Brisbane Powerhouse)

Image from gtaust.com
As for why its like this, the following explanation seems the most plausible:

Waay back in the day, airports were simply 1/2 mile or 3/4 mile squares of flat grass-covered former farm field with a small terminal building in one corner. Planes landed &amp; took off on the grass, and parked in front of the building to load &amp; unload.
Given the reality of rain &amp; snow, the area in front of the building would get muddier faster thatn the rest of the field. Between plane traffic, fuel truck traffic, cargo wagon traffic &amp; passengers (&amp; horses), churning the surface to mud would be pretty much a sure thing.
A pretty obvious innovation would be to pave a more-or-less fan-shaped area in front of the building where the airplanes could park.
Which when viewed from above, would look like an apron

As for tarmac, it is simply a synonym for Asphalt, which was used in apron's construction.
Note: The historical term apron bears similarities to modern ramp, as in 'walk the ramp'