Preheat oven to 350 degrees F (175 degrees C). Grease 18 muffin cups or line with paper muffin liners.
Whisk wheat bran, four, baking soda, and salt together in a bowl. Beat eggs in another bowl. Add yogurt, sugar, molasses, and applesauce to eggs; whisk until smooth. Stir bran mixture and chopped raisins into yogurt mixture until no dry lumps remain. Spoon batter into prepared muffin cups.
Bake in the preheated oven until a toothpick inserted into the center of a muffin comes out clean, about 20 minutes. Cool in the pan for 10 minutes before removing to a wire rack to cool completely.