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.