Q: How to transfer content from one iPhone to another using iTunes Mac Backup

My old iPhone (an iPhone 4s) is semi-working..it can be synched but the screen is black (so basically unusable). We can tell that it is on because we can hear a screenshot if we press the home and lock screen. We are mostly concerned about getting the contacts back. iCloud isn't an option because our family's iCloud has been messed up for months; we disabled it after we were receiving each other's contacts. I have successfully backed up my iPhone to my Mac via iTunes but am now trying to get it on to a different iPhone. How can I do this?