"Why?" is asking for a precise reason.
"How come?" means "Tell me about the circumstances which led to this situation."
For example, let's say some new neighbours move in. They're a blond, blue-eyed family, but you notice that the kids are speaking in Chinese. You think that this is strange, so you say to a friend who knows this family well, "How come they speak Chinese?". This means "Do you have any background information about this situation?"
As you can see, "How come?" isn't quite the same as "Why?". It's much less direct.