Why Smooth Surfaces?

I don't know any specific methods. I'm not a coin MD.

For ancient coins "smoothing" is often done to remove evidence of patches of corrosion. A smoothed field looks better preserved than it actually was. It is usually done with a small grinding tool, or even a dental drill.

The term could also be applied to the act of putting a completely smooth surface on one side or the other. This may have been done to prepare the coin for engraving, as a love token or some similar repurposing. Such grinding flat is often done with a sander or lathe.

Finally, a similar effect can be given to a coin through "natural" use. The English pub gambling game of shove ha'penny, for example, is played with a halfpenny coin. Using the same coin on the game board over several decades will wear one side of the coin down until it is worn smooth.

Tooling and smoothing are just ways of making a bad problem not quite disappear. I would hazard a guess of about 90%+ of coins treated in that manner can be identified as such.

Changing the mint mark, or lack of one, or date is a much bigger concern for the average collector.