I'll rephrase the sentence slightly to indicate what's wrong with option A (no change):
The designer envisions the game’s fundamental elements: (1) the settings, (2) characters, and (3) plots that make each game unique, (4) and is thus a primary creative force behind a video game.
When you use a colon, you provide a "definition" (or clarification) of something. In this case, it's for the game's fundamental elements. But the final part of the sentence has become disconnected from the main clause because it's been added as one of four list items—which makes no sense.
Option B inserts an em dash before the final part of the sentence:
The designer envisions the game’s fundamental elements: (1) the settings, (2) characters, and (3) plots that make each game unique—and is thus a primary creative force behind a video game.
This makes the final part of the sentence a continuation of the third list item, which also makes no sense.
Option C replaces the colon with an em dash, and inserts another as in option B:
The designer envisions the game’s fundamental elements—the settings, characters, and plots that make each game unique—and is thus a primary creative force behind a video game.
This version works. By using paired em dashes, the final part of the sentence is no longer part of the list items but is joined with a conjunction to the first part of the sentence.
Option D replaced the colon with a semicolon and inserts a semicolon before the final part of the sentence:
[The designer envisions the game’s fundamental elements]; [the settings, characters, and plots that make each game unique]; [and is thus a primary creative force behind a video game].
This results in a clause followed by two sentence fragments that make no sense.