Ahh, I see now. That was a much better explanation! So, I came up with this just then:

"Он читал книгу мне," = "He read the/a book to me."

Книгу is in the accusative case because it is directly affected by the first person reading it, and мне is in the dative case because the second person is reaping the benefits of what was being read, in a way, by listening. I think I'm beginning to understand this a lot better now!