As a breeder you tend to know your birds and can pick up on anything even slightly different with your birds. The birds aren't flying around loose they are confined to cages that are identical except for the size. They are different sizes depending on the size of the birds. You don't show birds that aren't in the peak of health. There would be no point as they will not display themselves off to their best. The whole idea is to win. The judges also go through an extensive training period. Trainee's have to judge at a large number of shows assisting the qualified judge. They have to study birds for their conformation. Traineeship is 12 months at least and to even qualify to train as a judge you have to be a breeder of at least 5 years standing, with budgies it is even longer. They then become probationary judges and can judge smaller shows on their own. They also have to pass tests before they qualify as a judge.
We also train our birds in the show cages so that they don't panic with many people around them, and so that they are comfortable in the cage and display themselves well. Some of the bigger shows have an avian vet present and any bird that is slightly suspect is removed from the show bench immediately. Believe me it is exceedingly rare for any bird to get sick either during a show or after.