It's important first to state that we don't know why Shakespeare made Othello a Moor. While we have much documentary evidence that Shakespeare was indeed Shakespeare, this is primarily in the form of legal and business documents. He didn't leave a journal or any explanation of his thought processes when it came to writing his plays. Therefore, we have to look at the text of the play for an answer and then make informed speculations.
We know that Shakespeare had to make it plausible that a strong, respected leader like Othello would be insecure enough to believe Iago's lies and manipulations about Desdemona. Shakespeare does that in part by making Othello older than Desdemona. However, marriages between older men and younger women were common in that era. It is likely that Shakespeare wanted to give Othello an additional flaw, particularly a cosmetic or outer one, that would make it easy for him to doubt that Desdemona could really love him.
Making him a dark-skinned Moor would make him an "other" in Venetian culture. We know that there is a prejudice against blacks in Venetian culture because of what Brabantio, Desdemona's father says to Othello:
Damn'd as thou art, thou hast enchanted her . . .
In those days, the devil was seen as black, and Brabantio is buying into stereotypes that Othello is "damned" because of his skin color. As if that were not enough, Brabantio says he would never, on his own, have allowed Desdemona to marry a black man. As he puts it, he would never have given her
to the sooty bosom
of such a thing as thou . . .
This is not likely to encourage Othello's confidence. Even without Iago, it might well have played on Othello's subconscious mind that the thoughts of the father might emerge in the daughter. Othello himself later blames his blackness for making him rougher and cruder than others, saying:
I am black
And have not those soft parts of conversation
That chamberers have . . .
That he feels shame over his skin color becomes apparent after he has decided Desdemona has betrayed him:
Her name, that was as fresh
As Dian's visage, is now begrimed and black
As mine own face.
Shakespeare, in making Othello black, hit on a way to make his insecurity and vulnerability about being worthy of Desdemona plausible. He is targeted in an evil and ugly way because his skin color makes him "other," and by internalizing that sense that he is "begrimed," he projects his own self-loathing onto Desdemona.