Simply put, it's like the restoration of an old art masterpiece, but instead of an image, it's the sound that the restoration artist works on.
Old phonograph recordings, particularly "soft cuts" or "instantaneous" recordings deteriorate with age and repeated playing. They acquire clicks, pops, crackle and surface noise that mask the recorded material, be it voice or music.
The goal of Audio Restoration is to reach through this curtain of unwanted noise and hear the recording like it was when it was first made. Think of a film of dirt on a mirror... the image is dull, blurred, flat and lifeless... but clean and polish that mirror and you can now see the image clearly.
Audio Restoration is designed to "clean up" a recording, much like the paintings of the old masters that have been restored to their original quality. The object is to separate the sound from the medium that carries it, so you can hear the music without the clicks, pops, scratches and hiss that often plagued the original recordings. In some cases, the desired sound is so close in character to the noise, that some noise has to be left so as not to sacrifice parts of the sound that we want to keep.
The person guiding the restoration process has to make many judgement calls along the road to clearer sound, since over-doing it can leave undesired low level artifacts of the process, often described as "thumps" or "Star-Warsy" metallic sounds.
Sometimes it's a balancing act to decide whether the artifact, when they occur, is more desirable than the noise it eliminated! Computers play a big part allowing us to do things that couldn't be done only a few short years ago. Mostly, we think that your ears will approve!