At its best, sports can help to bring a sense of universal connection to nations that see its teams compete. I think of the World Cup as one such example. Every four years, many people's attention from different nations converge towards the nation's soccer team competition. Promoting World Peace might be a rather lofty goal, but the idea of nations being able to compete without conflict and war, but rather through athleticism and sportsmanship could be feasible and one seen through events such as the World Cup of Soccer. The Olympics can also garner this level of competition and focus, but, again, the notion of world peace might be a bit lofty.
I don't disagree with the answer above, but let me play devil's advocate just so you have another point of view to consider.
With the Olympics, we are talking about world-inclusive set of athletic contests that happen every two years, winter and summer games alternating. Every country can participate if they wish. Every country is validated in opening and closing ceremonies. They are broadcast worldwide, watched by billions of people, and are a set of contests resolved by peaceful competition. While this doesn't directly cause all countries to be at peace, it is a peaceful event, a positive event, an egalitarian (based on equal opportunity) event, and certainly creates an atmosphere for peaceful relations. During the Cold War, it was one of the few ways in which Americans and Soviets interacted.
With World Cup soccer matches, this is a huge event in many countries, and growing in popularity in the United States. While many of the fans themselves have been violent at times, most of that is based on alcohol rather than true tendencies towards violence, with some notable exceptions. I think anytime countries come together to compete, it is overall a positive experience.
In my opinion, sports cannot really contribute much to peace in the world. In fact, I would argue that there are times when sports can actually hurt.
For example, tensions between Egypt and Algeria got to be quite high following the World Cup soccer qualifying match between the two countries recently. There was also the brief war between El Salvador and Hounduras (the Soccer War) in 1969. Neither of these was really caused by sports, but what I am saying is that when there is tension between two countries, having them play sports against each other can inflame the tensions.
Playing sports against people from other countries probably helps the actual athletes be more tolerant. But I doubt that it helps the general populations of the countries. They just get more upset when their team loses and they have another reason to hate the other country (S. Korea and the American skater Apollo Ohno, for example).