Sport is only the most obvious example. Why doesn't everyone just record and watch the game at their leisure? Or just get the highlights? Sure, those are options, but the enduring appeal of live sport points to there being more to it than just consumption of information.
Periscope for ESPN and watch the hosts betweeen quarters of some football game.
Periscope for famous people and watch them go on stage or whatever.
Periscope for you and let your family and friends meet your newborn asap.
A very large number of people streaming might create a curation requirement, but that's all. You have someone reputable who is retweeting or directing people to the most useful feed (as happens already with breaking news events on Twitter).
You end up watching someone having a dialog with a police officer, or in the middle of a rush, or something. You feel a greater sense of connection knowing that it is happening right now.