What does looking razor sharp mean?

Same thing as just "looking sharp." It means you look nice, attractive, classy...

ANS2:Since the adjective "sharp" is metaphorically applied to people who are dressed well or who are particularly intelligent, the addition of an object that describes just how sharp the person is would be a natural extension of the metaphor.

"Sharp as a razor" would mean very smart or good looking. You might also hear "Sharp as a marble". As you can imagine, that is not particularly complimentary.