I want to share a text like this app does
How can I do that?
There are two steps:
If you want your platform-independent Libgdx code to be able to call into Android-specific libraries, you generally need to create a small interface or bridge between the platform-independent bits and the platform-specific bits. See https://github.com/libgdx/libgdx/wiki/Interfacing-with-platform-specific-code
Once you're in the platform code, you need to invoke the correct Android Intent API. You did not provide any details about what kind of data you are trying to share (which may change which API you want to use), but see http://developer.android.com/training/sharing/send.html
You can spawn native Android processes from libGDX. Just spawn a share intent: http://developer.android.com/training/sharing/shareaction.html