The short and sweet answer is that you'll need to teach Flash to a Pokémon. Yup, you read that right. You'll need to have this subpar move on one of your precious Pokémon at certain points, despite HMs no longer needing to be taught anymore.
Certain spots in the game will be nearly pitch-back, allowing you to see only your own character and a small ring of light around them. To be able to see properly, you'll need to use Flash. You can also just stumble around the area for hours until you find your way (as a certain writer did as a child), but that is not recommended.
Why and how
Only the makers of the game themselves know why they decided to reimplement Flash as a semi-necessary feature or why it needed to be separate from all of the HMs on the HM Poketch app. But we can actually answer the how part.
First, you'll need a Flash TM. You can find one on the bottom floor of the Oreburgh Gate (the cave on Route 203), or you can buy one from the Veilstone Department Store. Then you'll need a Pokémon to teach it to. A few options are Psyduck, Pikachu, and Clefairy.
Once you've taught Flash to one of your Pokémon, enter the dark area. Open up your menu, select Pokémon, and select the Pokémon you taught Flash to. You'll find that Flash is now an option in bright blue text, right under Check Summary. Select Flash and your path shall be illuminated.
For more Pokémon Brilliant Diamond and Shining Pearl content, check out Turtwig, Chimchar, and Piplup evolution levels in Pokémon Brilliant Diamond and Shining Pearl on Pro Game Guides.