Q. Where (and why) does the Torah say that men must grow beards?
This law may have been designed to differentiate Israel from other peoples and pagan cults in which shaving was required or customary. The Talmud regards the beard as a symbol of male dignity, and hence it was a mark of shame to have one’s beard removed or shaved.
Some rabbinic authorities insisted that men should grow beards and not shave at all. Others interpreted the law as applying to the means of shaving and not to shaving as a whole, as what the strict law prohibits is cutting the corners of the beard by means of an instrument with one cutting edge.
We have portraits of clean-shaven rabbis from about the 17th century; they must have used scissors or hair-removing creams. These days orthodox men who shave use electric razors, though not if there is a blade that comes into direct contact with the face.
Some authorities say that the permission to shave applies outside Israel, presumably because Jews there constantly mix in general society and should not appear unkempt, but there are a number of leading Yeshivot in Israel where the students are clean-shaven and many modern orthodox Israelis are without beards.