Early in the war only a witness was required, but later they also required an actual wreck. As their war situation in the East deteriorated it became increasingly difficult for Luftwaffe pilots to get their claims verified. Over the Reich is was much easier as air defense units/observers usually could pinpoint where Allied aircraft went down (these were also salvaged and recycled for their metal).
The point of the system was to ensure that no more victories were awarded than wrecks found and documented. After the BoB German military intelligence became wary of overestimating enemy losses.
Oh, and it could take months, more then a year in some cases, for a German pilot to get his claims approved through the system. The Abschusskommission, which received all reports on crashed aircraft remains found by search units checked conflicting claims between AAA batteries and fighter pilots, and awarded credit for the victory to one claimant or the other. This system ensured that no more credits would be awarded than wrecks found.