Diane Levin posts about the use of food to aid in mediation:

quite a few mediators I know make a point of feeding their clients, not only because providing coffee and pastry is a nice thing to do, but for other reasons as well. Consider that the act of breaking bread with another human being is deeply symbolic, a reminder of the link between each person at table. (Note that the English word "companion" is derived from the Latin words for "together" and "bread", signifying "the one with whom I share bread".)

This makes sense to me as well. The act of eating is social and removes one from the immediacy of the situation being mitigated. Rather than focusing on the problem at hand as a problem - there is a greater chance that participants will see their rivals as people. In this case, people who like ribs.