First of all, we must realize that not everyone believes that traditional families are all that important to society. There are many who would argue that single parents, parents who have remarried, and same-sex couples can all be good parents.
That said, we can argue that traditional (or at least stable) families are important to society. The reason for this is that society has a tremendous stake in the raising of children. It is a cliché, but children are the future and a society that does not raise children well endangers itself. It is at least possible to argue that children do better when they grow up in a traditional family with both of their biological parents. They need to have both a male and a female role model so that they can understand what men and women are supposed to act like. They need to have their own parents so that they do not have to worry about whether stepmom or stepdad really loves them. They need to not have parents who are separated and who parent them differently and/or compete for their emotions. For these reasons, a traditional family is (arguably) best.
Traditional families, then, are arguably important for society because they do the best job of raising emotionally well-adjusted children who can function well in society.