Some Lua features can make it more pleasant to write the config files by hand. For example, comments (!) and local variables.
More powerful features can also be helpful if you have a large config file that would typically need to be generated by a separate program. One example of this is shown in the linked blog post, when they use a for loop to initialize a large array that has many similar objects.
It can also help if the config file starts gaining special "PRAGMA" fields over the years. There is a certain tendency for configuration files to slowly evolve into an ad-hoc and poorly implemented scripting language and in those cases, switching to a proper programming language cuts the problem at the root.
That said, for data exchange situations JSON has the advantage that it is more well-behaved in the case of malicious inputs.