The best thing about LibGDX is how complete and comprehensive it's API is. It covers OpenGL, OpenGL ES, Bullet Physics, Box2D and Freetype with a coherent API that's completely platform independent. It's platform support is also very good, being compatible with Apple iOS, Android, The Linux/Windows/Mac combo and even HTML 5. It also makes use of the best features of the Java programming language, allowing for very concise programs on the developer user side (all the typical Java verbosity is hidden behind LibGDX's API).
The other best thing about LibGDX are it's own libraries for Entity-Component development, common AI patterns, particle systems, value interpolation and game pad support.
It is also very easy to set up for Eclipse, NetBeans or IntelliJ development using it's absurdly simple setup tool. And if you don't like Java, LibGDX is also 100% compatible with other JVM languages like Scala.
And last but not least: it's free software both as in free speech and free beer!
That said, there is one not-so-terrible problem with LibGDX. It's documentation is sort of incomplete when it comes to advanced 3D programming, which it is perfectly capable of doing. The most affected part of this are the 3D animation and AI API documentations. The API and it's included javadocs are comprehensive enough that you can get around with it but some complete tutorials on the mentioned API's would be great.
The basic 2D API of LibGDX is simple enough to be used with only the tutorials from LibGDX's wiki. More advanced features like the Entity-Component API, the 3D API and the AI patterns require a bit more practice so prototyping some training applications will be required.
I Have used LibGDX to develop tutorial apps for local game design courses in my University, and I used it as the main development framework for my undergraduate thesis in Computer Graphics, developing an Augmented Reality control device for mobile robots as can be seen here https://www.youtube.com/watch?v=S_mK-1KjqG4 (video in spanish).