Preheat the oven to 375°F (190°C). Line a 12-cup muffin tin with paper liners or grease the cups lightly.
In a medium bowl, combine the wheat bran and buttermilk. Let the mixture sit for 10 minutes to soften the bran.
In a large bowl, whisk together the vegetable oil, egg, brown sugar, and vanilla extract until smooth.
Add the soaked bran mixture to the wet ingredients and stir to combine.
In a separate bowl, sift together the all-purpose flour, baking soda, baking powder, salt, and ground cinnamon.
Gradually add the dry ingredients to the wet ingredients, stirring gently until just combined. Do not overmix.
Fold in the finely chopped apple evenly through the batter.
Divide the batter evenly among the 12 muffin cups, filling each about 3/4 full.
Bake in the preheated oven for 22 to 25 minutes, or until a toothpick inserted into the center of a muffin comes out clean.
Remove the muffins from the oven and allow them to cool in the pan for 5 minutes.
Transfer the muffins to a wire rack to cool completely before serving.