INSTALL_ERROR_DISTRIBUTION_SIGNED_BY_APPLE

Here is how to upgrade from OS X 10.7.5 to a newer version of OS X — if your particular Mac supports that. OS X 10.7.5 is the end of that series of releases and there's no 10.7.6 available from Software Update (nor is that release likely), so you'll have to use the Mac App Store (from the menu bar, use  > App Store) to download a newer version — the current version is El Capitan 10.11 (10.11.4, right now), which is what you'll get from the App Store when you download and install El Capitan. Again, if your Mac supports 10.11.

In addition to the information in the link:

Your Mac needs:

OS X v10.6.8 or OS X Lion already installed

2 GB or more of memory (More is better - 4 GB minimum seems to be the consensus)

8 GB or more of available space

Check to make sure your applications are compatible.

Application Compatibility

Applications Compatibility (2)

El Capitan 10.11 Compatibility information

Do a backup before installing, preferable 2 backups on 2 different drives.