How does the Koran make the case for itself as the actual words of God?
This is a good question. There are several ways in which the Koran makes the case that it is the actual words of God. However, with this said, it is really based on faith. In other words, what we have here is circular reasoning - the Koran says it is the words of God; so, it must be. For example, it says in 4:82 these words:
Do they not consider the Qur'an (with care)?
Had it been from other than Allah, they would
surely have found therein much discrepancy.
Second, there is the argument of prophecy. In other words, if there are prophecies that are fulfilled from a thousand years ago, then there must be something divine about the Koran. The predicting of the future shows that the Koran is from God. Many religious people use this argument from other religions.
Third, there is the argument from advanced knowledge. In other words, if the Koran has information that was too advanced for the time, it points in the direction that this knowledge was given by God.
Finally, there is an argument from experience. Many Muslims claim that the Koran resonates them them on a personal and experiential level. To them, it alone is enough proof of divine origin.