What does “fully serviced” mean?

The watch movement is removed from the case. The case (and bracelet if there is one) is ultrasonically cleaned. The crystal is polished or replaced depending on condition. The case is not polished as this usually undesirable to collectors and detrimental to the value of the watch.

The dial and hands are removed from the movement. These are generally not cleaned because a) it very difficult to clean a watch dial without damaging it, and b) many collectors like the look of vintage patina. So the only treatment they get is dust removal with air blower and Rodico (watchmaker's blue-tac). 

The movement is completely stripped down and all the parts are cleaned with solvents in the ultrasonic cleaner. Each part is then inspected for wear or damage, and cleaned by hand if necessary. Parts which are worn or damaged are replaced (either with a N.O.S. part, or if unavailable, then from a donor movement). The mainspring is replaced if it is at all out of shape. The watch is then re-assembled and lubricated. 6 different Moebius lubricants are used depending on the part. The watch is then regulated with a timing-machine software. (All watches have a regulator which speeds up or slows down the watch. The software listens to the watch via a microphone and calculates how many seconds per day it is gaining/losing. The regulator is tweaked until this figure is as close to zero as possible).

The dial and hands are re-attached and the movement put back in the case. The watch is then worn for 2 days - 1 week to test it. During this time it is usually regulated again, as a watch's rate is usually different when worn compared to when static. For a vintage watch being within 30 sec/day is good (most factory specs were about that). If the watch is an automatic then this testing period also ensures that the auto-winding mechanism is functioning correctly.