This happens when you are currently signed into Blizzard with a different Blizzard account.

To sign in to Blizzard with the correct account, visit the Blizzard page, sign out and back in again using your correct Blizzard account. After that, go to http://hsreplay.net and sign in by pressing the "LOG IN WITH BLIZZARD" button.

You can also go to HSReplay.net -> Settings -> Connected accounts to verify all the Battle.net accounts you'd like associated with your HSReplay.net account are there.