How does this quote by Mark Twain "A lie can travel halfway around the world while the truth is putting on its shoes" relate to Othello?

This quote relates perfectly to
Othello because much of the plot turns on people believing lies and
deceptions instead of the truth.

This quote from Mark Twain
is saying that this is exactly how things go in life.  People hear and repeat and
believe lies.  That is why the lie can "travel halfway around the world."  By contrast,
people seem to be less inclined to believe the truth.

In
this play, Iago uses lies over and over again to maneuver Othello to his doom.  Othello
and others are willing to believe these lies (the biggest of which is that Desdemona has
been unfaithful to him) but not the truth.  Because Othello is willing to believe the
lies, he kills Desdemona.

The plot of this play depends on
people believing lies rather than the truth.  That is exactly what happens in real life,
according the quote from Twain that you cite.