Before the update, Apple let you quickly set up two iPhones or new iPads if you put one next to the other, but it only transferred some data, like your Apple ID and Wi-Fi settings. Now, you can transfer pretty much everything, including messages, pictures, app settings and more. Apps, however, are still downloaded after you finish the initial setup process. If you remember restoring a backup from iTunes, it's kind of like that, except much easier and uses only your old iPhone.
This is a feature that's been available on Android for many years, and it's a good option for folks who haven't saved a recent iCloud backup, who don't have enough iCloud space to save a new one, or who simply want to move data between two devices without putting any sort of backup in the cloud at all.
That's it. Now just sit back while everything is transferred over to your new iPhone. And remember, you can still use an iCloud backup if you want to, just choose the option to "download from iCloud" at the start screen instead of "transfer from iPhone."