Through the software:
Grab the GitHub software and install it.
In the top left there is a big Plus icon with a down arrow. Click it.
Choose Clone.
Find the Repo you have access to.
Click it and click Clone Repository
Choose where to put it (you could put it in a project, for instance).
Now it should be on the left menu
Click it and then click Sync in the top right, wait for it to download everything
The Repo is now on your HD.
Committing:
Make changes or add new files inside that directory
Open GitHub software, click the Repo, look at the top where it says "## uncommitted changes"
Click uncomitted changes, now you have to 'create a commit' to send to the repo.
Type something in the Summary (label for the commit)
Press Commit to master
Press Sync
So to commit something you have to create a commit with the data locally, then push it to the cloud repo. I think you have access to it, so this should work fine - otherwise you need to make a pull request (basically you saying "Hey, I did some stuff, do you want it?") and have an owner accept it. ("Neat stuff, sure we'll accept it into the repo.")