Corrupted backup from iOS 8.3 to iOS 9 I just downloaded iOS 9, and I get a message from iTunes saying that my backup is corrupted or not compatible with my iPhone.

Do you guys know why I get this? I tried to restart my computer, upgrade Yosemite (I don't want to install El Capitan for now), and restart my iPhone.
Not sure about your corrupted backup. But you can downgrade to iOS 8.3; the same way you upgraded to iOS 9. 
Your backup might show up if you do that. I upgraded to iOS 9. Couldn't find my backup. Downgraded to iOS 8 and it was there. 

If you need help downgrading, this should have the steps you need. http://www.macworld.com/article/2687162/how-to-downgrade.html

Find the correct 8.3 build for your phone here: https://developer.apple.com/ios/download/

In iTunes with your phone plugged in, hold the option key and press restore. Navigate to your downloaded build of 8.3 and select that. Restore to iOS 8.3
During setup, you should see the option to choose an iCloud backup if you have one.