The issues are similar to those that have occurred yearly with each new iteration of the iPhone. If this overload is like the ones in the past, the issue will solve itself as users update their devices over time.
There are more people simultaneously trying to reach Apple servers than the servers have the capacity for. Users have to connect to the servers to download the software, and again to verify the hardware and software.
With users updating their software, they are also likely to update their previously-downloaded apps to upgrade compatibility, which puts a further load on the servers.
Some users have estimated download times of more than an hour. When there is less demand for the servers, the issue will likely diminish. Until then, all there is to do is wait.