One definition of levigate is "polish". Mirrors are polished to make the reflection clearer and more accurate. We were created in the image of G-d to reflect His glory (Genesis 1:26-27). It seems as though He polishes our lives and removes the dust and smudges to make us better reflect who He is.
Telephone is a game where a line of children whisper a phrase from one person to the next. Each person must repeat the phrase exactly as they heard it spoken by the previous person. Obviously, mistakes and distortions are introduced along the chain as the phrase is heard and then repeated. By the time the phrase reaches the last person on the chain, it will have changed substantially from the original. The object of the game is to show how gossip results in distortions and inaccuracies.
As the telephone game was explained to the Hebrew children, they were confused at first: the children couldn't understand the point of the exercise.