TL/DR: Your operating system runs in a restricted environment to protect your computer. Running as administrator allows the PIA application to change your network configuration in order to protect your privacy and security.
Best practice for any operating system is to normally log on as a user with restricted privileges. This prevents you from accidentally deleting important files. If you need to do something that requires an administrator like change network configurations, rather than logging off and logging on as an administrator you can just select "run as administrator" (and supply the appropriate password if needed).
The Private Internet Access application requires you run as administrator due to the various options within the user interface. As a number of options change your network settings, such as IPv6 Leak Protection, DNS Leak Protection and the Kill Switch, the PIA application requires to be run in an elevated state to ensure that your privacy and security are maintained.
If you are concerned about running a VPN network in an elevated state, OpenVPN Portable and PortableVPN (which requires a paid license) are both portable apps which can be run in a restricted environment, however Private Internet Access do not support either application.