The Parsha begins with Hashem telling Moshe to speak to the Kohanim and tell them not to become טמא, defiled, through contact with a dead person, with the exceptions of their own immediate relatives (ויקרא כ"א:א'-ד'). Then the Torah says that a Kohein should not make a bald spot on his head, should not shave his beard, and should not make any scratches in his flesh (שם פסוק ה'). These last requirements are not unique to the Kohanim, since regular Jews also may not engage in these activities. Rashi (שם בד"ה לא ובד"ה ופאת ובד"ה ובבשרם) explains why the commandments are presented here, noting that each phrase teaches something new.
As for why these activities are forbidden, the Gemara in Kiddushin (דף ל"ו.) states that as a form of mourning for dead people, many non-Jews used to shave their heads. The Torah therefore comes to tell us that the Jewish people should not do that, and, as explained by Rashi in the Chumash (שם בד"ה לא), we may conclude that this is the reason not only for the Kohanim to avoid this practice, but for all Jews. We thus see that the Torah wants all members of Bnai Yisrael to refrain from imitating the non-Jews in this regard, and not just the Kohanim. In a similar vein, we read in our Posuk that it is prohibited for a Kohein to make scratches in his skin; earlier in the Torah (שם י"ט:כ"ח), that commandment is given to all Jews, and Rashi there (שם בד"ה ושרט) says that this is prohibited because such was the practice of the non-Jews as a sign of mourning. We therefore see again that no Jew, whether he is a Kohein or not, may follow this practice adopted by non-Jews and thus none of us are permitted to make a scratch in our bodies. As for the restrictions concerning cutting one's hair and shaving, they do not appear to have anything to do with non-Jewish mourning practices, but some Meforshim say that they too are designed to prevent Jews from looking like non-Jews.
We see from this Posuk that the Torah teaches us to take care of the way we dress and look. It is evidently important for us Jews not only to act differently than others, but to look different as well. Our style of dress and our personal grooming must also reflect the fact that we are Hashem's chosen nation. This does not mean that there is necessarily anything objectively bad about the way other people dress and look, it simply means that we as Jews must follow the directives of the Torah, and not the modern styles, in order to stand out as Hashem's people. If we look the way Hashem wants us to look and act the way He wants us to act, we will be publicly sanctifying His name.