That's what I thought. The problem is, You are using latest iOS 10 on your iPhone but an old iTunes/iPhoto on your computer. You need iTunes 12.5 or later to work with iOS 10 devices with Mac and other applications.
The latest iTunes is 12.5 and it requires OS X 10.9 or later. You need to upgrade your Mac OS X 10.7 to latest version. The latest is 10.12 Sierra. This is your only way to fix this problem. The problem you are having is incompatibility issue with Mac and iPhone version and there is no other way to make it work.
Follow the steps below to upgrade the OS X if your Mac support it.
Click on the Apple icon and App Store. This will bring App Store. Click Updates tab. If your Mac can handle latest OS X, then you will see macOS Sierra there with FREE UPGRADE button. Click Free Upgrade and enter your Apple ID and password.
This will download the update. It may take a few hours to download. Once the download is completed, it will bring the OS X Installer. Follow the onscreen instructions to install it. It may take up to 3 hours to install the upgrade.
Upgrading the OS X will also give you a new application named Photos app. Photos is a replacement of iPhoto (Apple stopped updating iPhoto) and it does allow you to access your old photos. So you have to download using Photos app and iPhoto will no longer work on 10.12 Sierra.
If you have any questions or would like any additional information, please don't hesitate to ask.
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