To bend the point of a nail after it is driven home. The word is sometimes written clench, from the French clenche, the lift of a latch. (German, klinke; Dutch, klinken, to rivet.) (See page 261, col. 1, Clench)
A lie is called a clincher from the tale about two swaggerers, one of whom said, “I drove a nail right through the moon.” “Yes,” said the other, “I remember it well, for I went the other side and clinched it.” The French say, Je lui ai bien rivé son clou (I have clinched his nail for him).
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894