The link below gives a really good and detailed account of the reasons why Yugoslavia split. Please consult it.
The basic reason why Yugoslavia split is that there were too many different ethnic and religious groups within the country. As an example, Serbs and Croats are ethnically somewhat different. They write their language with different alphabets. Most importantly, Serbs are Orthodox Christians while Croats are Catholic. These differences made the groups feel different from one another. (And there many other ethnic groups as well...)
The differences between these ethnic groups were not a problem during communist times because the communists kept the country under tight control and conflict could not erupt. Once the communists left power, however, the various ethnic groups were free to act as they wished. This led to conflict between the groups and those conflicts eventually led to a horrible civil war with terrible atrocities committed by a variety of sides.
So Yugoslavia split because it was made up of groups that did not see themselves as similar to one another. Once the repressive communist government was gone, these groups came into conflict and the country split apart.