Watch the World Burn has two primary uses: trap disarming (this removes the trap from the game before it can fire) and countering asset spam (avoid trash punishment/cost.)
The first is exemplified versus Jinteki Biotech: Life Imagined that's running Cerebral Overwriters, Mushin No Shin, and 5/3 agendas. If they Mushin out a card and advance it, you're usually stuck with gambling on whether it's a trap or an agenda. Watch the World Burn eliminates that threat, letting you run it without fear. But not only does it remove the trap you access from the game, it also makes you immune to that trap for the rest of the game.
The second is obvious against NBN: Controlling the Message, avoiding the trace, and getting a discount on trashing expensive cards. Being able to do it on all subsequent copies lets you cripple a part of their gameplan. Just beware of passing through a Data Raven on your last click (this event is a terminal, so you won't be able to clear tags afterward.)
Even if the card you remove has a trash cost of 5, you still only save 2 credits, which is a bad return on investment. So unless the remove-from-game ability or the all-copy removal are relevant, as it is in the above scenarios, it's not worth using.