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Short version: The rule is about appearing nice, clean-cut, and respectable in line with current cultural norms. It's a matter of practicality - just look nice, in order to better represent the Church and therefore, God.
From an address President Dallin H. Oaks gave to the 25,000 students of Brigham Young University.
Unlike modesty, which is an eternal value in the sense of rightness or wrongness in the eyes of God, our rules against beards and long hair are contemporary and pragmatic. They are responsive to conditions and attitudes in our own society at this particular point in time. Historical precedents are worthless in this area. The rules are subject to change, and I would be surprised if they were not changed at some time in the future. But the rules are with us now, and it is therefore important to understand the reasoning behind them.
In the minds of most people at this time, the beard and long hair are associated with protest, revolution, and rebellion against authority. They are also symbols of the hippie and drug culture. Persons who wear beards or long hair, whether they desire it or not, may identify themselves with or emulate and honor the drug culture or the extreme practices of those who have made slovenly appearance a badge of protest and dissent. In addition, unkemptnessβwhich is often (though not always) associated with beards and long hairβis a mark of indifference toward the best in life.
Upvote:2
I currently serve as a Bishop in the Church of Jesus Christ of Latter-Day Saints. I wore a beard before I was called to serve in this capacity. I was not required to, nor was I asked to shave. I shaved none the less as it seemed like the appropriate thing to do.
I know of only 2 instances where the male members are required to be clean shaven. As a full time Missionary for the Church and as an Ordinance worker in a Temple. It has definitely been my experience that most leaders in the Church are clean shaven, although there is no official requirement to be so, at least for Stake Leadership and local Ward Leadership. Being the norm in Leadership positions in the Church, I can only assume that being clean shaven stems from a desire to follow the example of those we believe to be Prophets and Apostles.