My boss set up a repo in our shared GitHub and I want to add a description, but there is no edit button to do so:
I am the only contributor so I want to add one to make it easier to find. How can I do this?
Since there is no "Edit" button or "Settings" tab visible in the screenshot, it means you're not an administrator of the repository.
You'll have to ask your boss to promote you to Admin status on the repository, or ask them to set these up for you.