After several months of dealing with iOS9's new way of "backing up" locally, I find that this is the simplest explanation for those who are STILL confused about how things work and how to maintain your "plug in to your computer on a weekly basis" backup schedule.
When you plug in an iOS9 device to iTunes and hit "Sync" or "Backup Device" iTunes is taking a snapshot of what download Apps you have on your device and where they go/how you've arranged them, along with other things like WiFi connections, Music, Books, Videos, etc. It's not actually PHYSICALLY transferring that data to/from your computer any more, it's just making note of what you had on there and where it was.
If you want to be able to "restore" from that backup you need to ensure that everything that was on your phone is on your computer and in iTunes.
The best way to accomplish that is to go your iTunes Preferences>Store and ensure that "Automatic Downloads" are on for everything that you want to maintain backups of from your phone are checked off.
Second, go to your account (the little silhouette icon in the top right of iTunes) go to "Purchased" and DOWNLOAD ALL of the apps/music/movies that you have in Apple's store records/cloud. Have it ALL on your iTunes for any potential "backup" configuration you make, otherwise when you attempt to restore from backup, if it's NOT on your computer it will no longer restore properly.
That's it. I've successfully restored full phone backups via this method and literally the ONLY thing that doesn't restore is Bluetooth profiles, which makes sense because those are tied to unique devices.
There's really no other way to do it by plugging in and backing up on iTunes. You can alternately turn on iCloud for Apps, Music, Videos, etc. and have everything backup/restore that way, but I find the amount of time that that method takes to download 30 Gigs of apps is pretty maddening and and sometimes no matter what I do, some apps don't ever properly download from the cloud. *shrug*