Background syncing was added recently, and it seems to have introduced some confusion, so I will be making some tweaks in the next update. In the mean time I’ve updated the above link to add the following:
If you have a lot of media, the initial sync may take quite a while, and if you leave your device unattented, the screen may turn off, interrupting the sync. To prevent this from happening, you can go into iOS’s settings screen, and temporarily turn off auto lock in the display preferences. Once syncing is complete, you can turn it back on again. Future AnkiMobile versions will likely do this automatically for you when the syncing screen is visible.