This is a very interesting question. Reading between the lines, I find you are asking us about life today, ... not life back in Shakespeare's time. Why do "we" find the issue of race relevant? We find the issue of race relevant because racism exists today just as absolutely as it did back then. I'm going to echo litteacher8 and go a bit further. Yes, we have different issues. Yes, slavery was part of the horrors of the United States back a couple of centuries ago. We are still healing. Let's even go deeper than the idea of an interracial marriage. So, seeing characters plant the seeds of suspicion about a black man, young and in love, ... with other characters that may already doubt him because of his race is very damning. This very thing is one of the many reasons why the plays of Shakespeare (and Othello is no exception) are always considered to be universal.
Racism existed then, just as it does now. The idea of a mixed race couple was also unpopular then. White men felt that they had first dibs on white ladies. They did not approve of black men dating their ladies. Plenty of racial slurs and jokes are used to describe Othello in the play.
I think that racism is used in Othello in order to accentuate the idea that Othello is an outsider. The use of race in the characterization of Othello is one where it only adds to his self perception that he is fundamentally different than all others. Othello recognizes that his is different. He is a warrior, as opposed to a politician. He is one who is only recently wealthy and powerful, and not someone that has had it in his lineage. Race is just another element to add to his difference. It is also another level towards which insecurity is felt. The fundamental challenge in Othello's characterization is one where he feels different than the rest of society because of his race. This helps to enhance his insecurity about why someone like Desdemona would be with him. The issue of race is brought out between both of them, with her being the epitome of whiteness and he of blackness. This visual difference has settled, to an extent, inside Othello and is something that Iago manipulates within him. The declaration of Othello as a "black ewe" is quite deliberate, something that Othello recognizes as fundamentally true in that he is unable to reconcile how his race would not be used against him by Desdemona. In this light, Shakespeare introduces the issue of race to heighten the difference that Othello experiences, and the insecurity and doubt that come along as a result.