Note that the discard is part of the effect, not a cost. So if you discard a "Dark World" monster whose effect activates when it is discarded from the hand to the Graveyard by a card effect, you'll get two benefits for the price of one.
In a "Dark World" deck, you can combine this card with "Evilswarm Nightmare", to stall your opponent when he/she Special Summon a monster, destroy that monster with this card, discard a "Dark World" monster and take profit of its effect.
You could use this combo with "Sangan", and get a free search in the process. (Traditional Format only).
This card can be useful if you think your opponent has a "Solemn Judgment" or other Counter Trap Card face-down, making suitable targets to destroy and then discard and activate the effect of your discarded "Dark World" monster. Your opponent would think twice to attempt countering this card.
Even in non-dark world decks, this can serve as a great card-removal card. Since the discard is part of the effect, you do not need to fear about having to pay the cost if this card is negated.