Why wouldn't my iPhone 6s plus transfer data from my back up in iTunes?

I backed up my old iPhone 5 to iTunes just before I left to buy my new iPhone 6s+. When I plugged my new iPhone in to my computer it said the phone need a software update before anything could happen and it would take 5 hours. When that was done I followed Apples directions to transfer my data from my iTunes back up but the new empty iPhone had been backed up to iTunes after the update. I did download some things from iTunes onto the phone directly but the all important phone contacts are missing. I am not tech savvy.

Are you saying it transferred everything but contacts?

By default, I believe apple stores contacts in iCloud. Sign into that and turn the Contacts switch to on. Now give it a few hours or so and then check. (It could happen instantly, in 5 minutes or in an hour. I've had varying results) if they aren't there, reboot. Hope it helps :)