Why did Shakespeare choose to set the play Othello in Venice?
Shakespeare wasn't, it seems, all that interested as a playwright, in inventing story and setting. He often borrowed (or downright stole) his plots from other authors, and by extension, the story's setting. Othello is no exception. So, the simplest answer is that this is how he found the story, so this is the way he wrote the story.
Certainly it could also play in to his thinking that the audience was much more intrigued with the more exotic locales of Italy and the island of Cyprus as places and people to "visit" while attending the performance. This idea is not far off, by the way, from the modern inclination (especially in movies) to have futuristic (The Matirx) or imaginary (Avatar) worlds as settings meant to entice large crowds of audience. Shakespeare, we can assume, created his plays with the same sorts of enticement in mind.
So Shakespeare, on one level, took the path of least resistance and left the setting as he found it in the borrowed plot. But on another level, he was very conscious of how intriguing it was for his audience members to see and experience locations that were very foreign and exotic to them.
As Shakespeareguru notes, Shakespeare based many of his plays on historical information. Interestingly, only Act 1 of Othello is set in Venice; Acts 2-5 are set on the island of Cyprus. This setting isextremely important because it facilitates the action of the play.
Between Acts 1 and 2, Othello, along with the other characters in the play, make the journey from Venice to Cyprus in anticipation with a naval conflict with the Turks. However, during that time, the Turkish fleet is shipwrecked in a storm. Thus, Othello and the others arrive on the island of Cyprus, and since there will be no war, they have nothing left to do except celebrate. (They celebrate Othello's marriage and the fact that the Turks have been shipwrecked).
Because the characters are away from their homes, and because they are relatively unable to just pick up and head back to Venice after such a long journey, Iago's pan is more effective. On the island of Cyprus (during the celebration on the first night), Cassio gets drunk, gets in a fight with Roderigo/Montano, and is stripped of his lieutenancy. Obviously, this incident facilitates Iago's plan.