It mostly has to do with finding nice compactifications. Compactifications of varieties are a good thing as they allow us to control what happens at "infinity". If the variety itself is smooth it seems a good idea (and it is!) to demand that the compactification also be smooth. However, you need the situation to be nice at infinity in order to make the study of asymptotic behaviour at infinity to be as easy as possible. The best behaviour at infinity would be if the complement were smooth but that is in general not possible. What is always possible is to demand that the complement be a divisor with normal crossings. In practice it works essentially as well as having a smooth complement: You have a bunch of smooth varieties intersecting in as nice a manner as possible.