I'm new to programming and I am trying to develop an html5 game. I am planning to use the libGDX framework to get the job done. I have tried downloading it on to my Mac but the app has not shown up. Does this have something to do with not having an Apple dev account?
As far as I know you don't need an Apple dev account (99$) as long as you are not going to develop your application for IOS.
You also need an IDE to develop your program on. I personally use Eclipse for Windows. But since you are using Mac, you would probably prefer XCode 7, which is free. You also need Java JDK and Java JRE. Then you can download the libgdx project creator from here.
I would strongly suggest that you read the documentation for libgdx.