This belief goes back to the early days of the Moroni statues. The very first statue on the Salt Lake temple faced east, as did the temple itself. But one occurrence of something does not make a rule. This tradition really begins with the second Angel Moroni being placed on the Los Angeles Temple.
Original plans for the temple had the Moroni facing in line with the front door, to the south east. After the statue had been placed, President David O. McKay, on site for the placement, asked that the statue be rotated 90 degrees to face north east. People assumed there must be a doctrinal reason for the move, though no reason was given and it is equally likely that it was an aesthetic one. The tradition of ‘Moroni must face east’ was born, despite the fact that Seattle Washington, the fourth temple to receive an angel statue, has both the temple and the statue facing west. For more information on which direction Angel Moroni faces, see this section.
More thoughts on East
The turning of one statue is not the only basis for this belief. East holds significance in scripture. Some temples have doors on the east side, which is symbolic of the east gate of the temple in Ezekial”s vision:
“And the glory of the LORD came into the house by the way of the gate whose prospect is toward the east.” – Ezekial 43:4
There are other references of the Lord appearing from the east and moving towards the west in the scriptures, as he is to do when he comes again.
“For as the lightning cometh out of the east, and shineth even unto the west; so shall also the coming of the Son of man be.” – Mathew 24:27
There is a symbolism in the movement. When Adam and Eve were removed from the Garden of Eden they were sent our eastward. This is movement into the east. Moving into the east became a symbol for falling, apostasy and wickedness. Having Moroni Face eastward then becomes more symbolic of the righteous in the West (Nearest the Garden of Eden, the first temple) preaching to the wicked world in the East of the Garden of Eden.
You will often find moving from east back into the west being used in the scriptures to represent renewal, return, restoration and repentance. A symbol of returning to the Garden or the Temple, and a symbol of the return of Christ.
Faces: North East by East