The ability triggers whenever any player (including you) casts a spell of any type.

This has the supertype world. When a world permanent enters, any world permanents that were already on the battlefield are put into their owners' graveyards. This is a state-based action called the "world rule." The new world permanent stays on the battlefield. If two world permanents enter at the same time, they're both put into their owners' graveyards.

This is not an additional cost, it is a triggered ability.