About a year ago, I asked a similar question in this subreddit.

This was /u/TectonicWafer's answer.

It originates in the OT, in Levitical law, where the men are instructed not to round off their beards or the hair on their heads. That's why Orthodox priests look very much like Orthodox Jews, who follow the same instruction. The typical interpretation is that it's an expression of ascetical disregard for mundane concerns (asceticism is huge in Orthodoxy). Orthodox mostly follow the same general ascetic practices, unlike in Catholicism, where monasticism and the priesthood evolved into many different orders each with their own rules.