This file contains a skeleton framework for using LibGDX and Ashley. You should be able to compile this either on command-line or using your chosen IDE.
See this for Eclipse, IntelliJ-IDEA and NetBeans installation details (you don't need Android/iOS stuff)
For compiling, see this and choose your IDE.
Upon finishing compilation you should be able to go to desktop/build/libs/desktop-1.0.jar and run it.
Check your Eclipse version! Go to Help->About Eclipse - If you don't have version 4.4, change 4.4 in the link to whatever version you have, or update to 4.4.
Note that in Eclipse, you can get the GWT from http://dl.google.com/eclipse/plugin/4.4
You need the Google Plugin for Eclipse and the Google Web Toolkit SDK (under SDKs), this lets you compile your Java code to Javascript and run it in the browser! In turn, this makes it a LOT easier to make preview and test versions that can be shared!