Okay, I am yet to see this feature my self, but Instagram Stories are now available in your web browser. The Instagram team just posted this update on their blog which also states that it could take some time before everyone can enjoy Stories outside the Instagram app.
With the stories feature now available on the web, you can view them on your mobile or desktop browser. Prior to this update, Instagram Stories were only available inside the Instagram app.
On a mobile browser, you will see fresh stories on the top of the feed. On a desktop browser, you will see them on the right. Needless to say that you need to log-in to your Instagram account first.
Currently, this update allows users to only view stories. But Instagram has suggested that the ability to upload stories from web browsers is coming soon. The upload feature will only be available on mobile browsers, though. As The Verge reports, Instagram has no plans to bring the ability to desktops.