With fat binaries you are looking to isolate your application from environment -- by providing the dependencies (library code) along with your base application code. This way you don't have to make sure the user of your application provides the correct libraries in correct versions.
Docker container is an extension of that idea. If you can distribute some of the libraries, why not distribute entire execution environment? Why not isolate your application from external environment even further?
The more you control the environment the less work you have to do to make sure the application will work.
My point of view, there are two types of applications with diametrally different development process:
1. applications where you have to invest a lot of effort into making it resistant to the environment -- it has to work regardless of the differences in user environment. Think for example in terms of a PC game that must run on different GPUs, with different driver versions, on different OS versions, with different applications installed.
2. applications where it is enough to show that there is a set of circumstances when the application works correctly -- think in terms of typical enterprise application where DEVs would place it on the server and then the environment would be religiously preserved to not risk application breaking.
Developing type 2 applications is much, much, much less effort.
By distributing type 1 applications along with the environment we basically reduce the cost of developing them because now it would be enough to develop them up to type 2 applications standard.
For example: you may create a simple bash script to do something. If you distribute it as just script file, you need to make the script work on every bash on every type of mainstream Linux distribution.
If you distribute the script as a docker image it is enough for you to make that script work on that particular image (for example Ubuntu x.x) and you don't need to worry about other possibilities.
This is the essence and value in distributing of applications as Docker images.