How does Package Disabler app work? There are two apps that can disable system apps (bloatware) from Samsung and LG devices. Those do not need root or any special permission. Can anyone explain how this works?
There is a difference between disabling and removing bloatwares. Disabling can be done without root, but removing them can be done with root only. 

The app for LG that you mentioned changes device ownership permissions. It asks you to make itself the device owner. Quoting this site,


  A Device Owner is an application that runs as a device administrator on your Android 5.0+ device. The Device Owner app can use the programming methods in the DevicePolicyManager class to take control of the configuration, security and other applications on the device. A device can have only one Device Owner. No other applications can utilize the methods in that class to control the device.


As a device owner, the app can modify the partitions that cannot be otherwise accessed without root. Basically you can consider this a restricted root access (restricted because root permissions have many more privileges compared to device ownership). 

Remember that making any app a device owner is kind of playing with your phone's security, so don't do that unless you trust the app fully. In root, you have an option to deny root access. Here, you won't even know what the app is doing under cover, and it might be late before you actually remove it.