A person [i.e., a priest] may examine anyone’s skin afflictions, except for his own. Rabbi Meir adds, nor [may he inspect] the afflictions of his relatives. A person [i.e., a sage] may release anyone from faulty vows, except for his own vows. Rabbi Yehudah adds, nor may he [release] the vows between his wife and others. A person [i.e., an animal specialist] may inspect anyone’s firstborn animals except for his own firstborn.
The three cases cited in this mishnah all describe situations in which an expert is disqualified from diagnosing his own situation. Negaim, or skin afflictions, are specific types of discolorations on a person or his possessions that can have major ritual and financial consequences. Nedarim, or vows, are promises that can sometimes be disqualified based on technical flaws in their execution. Bekhorot, or firstborn animals, must be turned over to the kohen unless they bear certain permanent blemishes, in which case they may be kept by the owner. Obviously the thematic connection is that an expert may not rule on cases affecting him.