Twilight is likely losing because she memorized only the Sicilian Defense, and it's variations, and no other opening. Also, knowing Twilight, she would be frustrated by Pinkie Pie making moves that aren't considered the best, and either focus too much on one thing or over thinking it.
Based on what I see, it appears that Pinkie Pie has four pawns, two rooks, and one knight. Twilight has two pawns and one knight. Given what I see, unless Pinkie makes blunder after blunder, Pinkie Pie will win. The reason is that in two turns, Pinkie will be able to move her pawn across the board and make it a Queen (I guess it's actually a Prince in this game, but I'm using the chess term Queen in this case). After Twilight moves her Princess up to attack the Knight, Pinkie could just attack Twilight's Knight with her pawn, and let Twilight take her Knight. After that, Pinkie will be able to move her pawn to the end of the board. Twilight's Princess is too far away to help, so there is nothing that Twilight can do to stop it.