I am a Cohen and I don’t shave with a razor, I know that is prohibited. I would like to know the reason for this prohibition. Not just the Torah rule but the underlining principle of why we cannot shave.

The prohibition to shave is on the whole Israeli nation, not just Cohanim. Sefer Hachinuch explains in Mitzva 253 that since idol worshipers customarily act as such, the Torah told us to keep away from their actions so that we don't resemble them in any way, even by things that aren't connected to their faith. Rabbi Ro'i Margalit