In this case they compare the human intellect (mental abilities) and a razor edge.
You extend the analogy when you "continue to use" the analogy and apply it to some of the attributes of the two things you are comparing.
Sometimes extending the analogy is not appropriate because although the analogy works, extending the analogy to specific attributes doesn't work.
In this example comparing the human intellect to a razor edge works because both are "sharp". But while the razor becomes blunt/dull the more you use it, the human intellect becomes sharper the more you use it.