To unlock this feature you must first complete a hidden challenge.
Obtain at least 3 copies of "Ultimate Blue Eyes White Dragon" from the very first box and you will be able to watch other people's replays using your account.
Yeah, for some reason duel replays can only be opened on a phone that has Duel Links installed, despite having an URL suggesting that it can be opened on PC as well.
Konami really should start figuring out a way to watch replays on PC.