To ensure that they have not completed the run with less weight than they started out with, either fraudulently or by accident. In handicap races, for example, weight is allocated to horses so that - in theory at least - all the runners should cross the finishing-line together. It never happens, of course. However, the extra weight over and above the jockey's own is provided, if necessary, by metal plates in their saddle-cloths. These might fall out and that would give that horse an unfair advantage. Hence, the weigh-in.