Here "peace" refers to inner peace. Alternate translation: "so that you may have inner peace because of your relationship with me" (See: [[rc://en/ta/man/translate/figs-explicit]])
Here "the world" refers to the troubles and persecution that believers will endure from those who oppose God. Alternate translation: "I have conquered the troubles of this world" (See: [[rc://en/ta/man/translate/figs-metonymy]])