The functionality is slightly different that what you've likely seen on your phone. Clicking a profile picture brings up the Story for that user. From there, you can click or use the arrow keys to scroll through, but there's no way to move on to the next post unless you exit out of what you're viewing. Right clicking on a profile image will allow you to download the story. Instagram hasn't always taken kindly to feed-reading apps and plug-ins that use its APIs, so this browser extension may not be around for long.
Since its introduction, Instagram has always taken a mobile-first approach. The app was available for a while before posts were viewable on the web. Even then, you couldn't take a gander at your feed in a browser until a few months later. The social photography app has embraced web users to a certain degree though, redesigning profiles and adding both search and notifications over the last year. Only time will tell if Instagram will make stories available in your browser in an official capacity.