A key point about the action moving to Cyprus, I believe, involves the issue of Othello's military prowess and his loyalty to the Venetians.
Venice, and Europe as a whole at the time Othello takes place, felt vulnerable to attacks by the Ottoman (Turkish) Empire. The relatively recent fall of Constantinople, one of the last major outposts of the remnant of the eastern Roman (or Byzantine) Empire, was a shock to Europe. Othello, in spite of his non-European background, is a major part of the Venetian (and by extension European) effort to block the Ottoman advance and to protect the European continent.
In the second act it is revealed that a naval battle has taken place, and that the Venetians have defeated the Turkish forces. Othello's military ability, demonstrated here, is important to the action of the play, partly because it makes him an ideal husband to Desdemona for every reason despite the racist objections to him that have been raised. This demonstrates the wrongness of the racist views held by Brabantio and exploited by Iago.
In some sense the play can be seen as a metaphor of the conflict between European and non-European cultures. The setting in Cyprus, an island in the eastern Mediterranean which partakes of both cultures, is emblematic of that conflict. Othello himself is tragically caught in the middle, as a man of non-European descent giving his loyalty and love to the Venetians and to Desdemona.